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An Introduction to the
Theory of L-functions
Jörn Steuding (Würzburg University)
A course given at Universidad Autónoma de Madrid, 2005/06
1.5
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Contents
Preface
iii
Chapter 1. The classical L-functions of Dirichlet, Riemann & co.
1.1. Motivation: prime numbers
1.2. Riemann’s zeta-function
1.3. Dirichlet L-functions
1.4. The prime number theorem
1.5. Tauberian theorems – a general approach
1.6. The explicit formula
1
1
7
14
24
35
49
Chapter 2. Zero-distribution of the Riemann zeta-function
2.1. The Riemann hypothesis
2.2. The approximate functional equation
2.3. Power moments
2.4. Hardy’s theorem: zeros on the critical line
2.5. Density theorems
2.6. Universality and self-similarity.
66
66
72
78
84
87
95
Chapter 3. Modular forms and Hecke theory
3.1. The functional equation for zeta and more
3.2. The zeta-function at the integers
3.3. Hamburger’s theorem
3.4. Modular forms
3.5. Hecke’s converse theorem
3.6. Shimura-Taniyama-Wiles
105
105
117
121
124
129
134
Chapter 4. The Selberg class – an axiomatic approach
4.1. Definition and first observations
4.2. The structure of the Selberg class
4.3. The Riemann–von Mangoldt formula
4.4. Primitivity and Selberg’s conjectures
4.5. Hecke L-functions
4.6. Artin L-functions and Artin’s conjecture
4.7. Langlands program
143
143
146
149
160
167
173
185
Bibliography
191
ii
Preface
This course provides an introduction to the theory of L-functions, a topic
which plays a central role in number theory since Dirichlet’s proof of the
prime number theorem in arithmetic progressions in 1837 and Riemann’s
famous path-breaking paper in 1859. L-functions are generating functions
formed out of local data associated with either an arithmetic object or with an
automorphic form. These functions are special examples of so-called Dirichlet
series; all of them have in common that besides their series representation
they can also be described by an Euler product, i.e., a product taken over
prime numbers. The famous Riemann zeta-function
−1
∞
Y
X
1
1
=
1− s
ζ(s) =
ns
p
p
n=1
may be regarded as the prototype. L-functions encode in their valuedistribution information about the underlying arithmetical or algebraic structure that is often not obtainable by elementary or algebraic methods, e.g. the
classical prime number theorem which states that the number π(x) of primes
p ≤ x is asymptotically equal to the integral logarithm, resp.
log x
= 1.
x
Another example is Dirichlet’s analytic class number formula which measures
the deviation from unique prime factorization in the ring of integers of quadratic number fields. Two of the seven millennium problems are questions
about L-functions: the famous Riemann hypothesis (all non-real zeros of ζ(s)
lie on the critical line Re s = 21 ) and the conjecture of Birch & SwinnertonDyer (the rank of the Mordell-Weil group of an elliptic curve is equal to the
order of the zero of the associated L-function LE (s) at s = 1).
lim π(x)
x→∞
We want to give an overview of the variety of L-functions, their importance
for number theory and allied fields, and recent progress toward old and new
problems. After introducing the classical examples, as ζ(s) and Dirichlet Lfunctions, and studying basic properties, we concentrate on three main lines
of investigation in detail. First, we give a rather detailed account on studies
of the zero-distribution of Riemann’s zeta-function. We shall prove that ζ(s)
has infinitely many zeros on the critical line and further that there cannot be
too many zeros off the critical line. This supports the Riemann hypothesis.
It is believed that a proof of the Riemann hypothesis for the zeta-function
should easily carry over to other L-functions and, indeed, most of the techniques in the second chapter can be generalized; however, these techniques
alone will probably not be sufficient for a proof of the Riemann hypothesis.
iii
Second, there is Hecke’s theory which links modular forms and Dirichlet series with functional equation (Wiles’ et al. proof of the Shimura-Taniyama
conjecture, including Fermat’s last theorem, marks one of the highlights of
this approach); here we shall meet further examples of L-functions and learn
new techniques going beyond the theory of the nineteenth century (or those
designed to deal with the zeta-function). Finally, we study the axiomatic
approach initiated by Selberg with its far-reaching consequences on many
number theoretical problems as, for example, Artin’s conjecture on the holomorphy of Artin L-functions subject to the truth of Selberg’s orthogonality
conjecture.
There is another quite remarkable line of investigation, namely the impact
of Random Matrix Theory, i.e., the recent idea to model L-functions by large
unitary random matrices; this approach is motivated by Montgomery’s celebrated pair correlation conjecture and computations observing that the nearest neighbour spacing for the nontrivial zeros of ζ(s) seems to be amazingly
close (statistically the same?) to those for the eigenangles of the Gaussian
Unitary Ensemble. These observation have restored some hope to an old idea
of Hilbert and Polya that the Riemann hypothesis follows from the existence
of a self-adjoint Hermitian operator whose spectrum of eigenvalues corresponds to the nontrivial zeros of the zeta-function. First it was our intention
to give a brief account of these ideas in the notes too; however, by lack of
time we did not include this approach here. We hope to add this approach
in a later version of these notes.
The course is aimed at doctoral students and non-experts which want to
learn the fundamentals of this subject. Of course, it is far beyond the scope
of this course to prove all relevant results, for instance, the rather technical
converse theorem of Weil (or the Shimura-Taniyama-Weil conjecture which
I hardly understand myself). However, we want to sketch the main ideas in
order to obtain a first impression on the theory of L-functions, to learn its
big picture-questions and the modern approaches with which these objects
are studied. These notes contain more material than that presented in the
classroom (where we had two hours per week); furthermore, we have added
many exercises (the advanced marked with an asterisk) with the aim to give
the interested reader the possibility to get in touch with the basic objects
and to practise the presented techniques.
I am very grateful to Fernando Chamizo, Keith Conrad, Ernesto Girondo,
Fernando Holgado, Rasa Steuding, and Adrian Ubis for valuable comments
and corrections.
Jörn Steuding, Madrid, January 2006.
iv
CHAPTER 1
The classical L-functions of Dirichlet, Riemann & co.
The main theme in this introductory chapter are prime numbers. Questions
about primes had been a driving force for number theory ever since their
discovery by the ancient Greeks. Prime number distribution is intimately
linked with analytic objects, so-called L-functions. In this first chapter we
will introduce some classical examples: the Riemann zeta-function, Dirichlet
L-functions, and Dedekind zeta-functions. The particular case of Riemann’s
zeta-function, the prototype of an L-function, will be discussed in detail. We
shall learn first fundamental properties, prove the celebrated prime number
theorem, and get to know the big open conjectures as, for example, the famous
Riemann hypothesis. For further reading we refer to Apostol [2], Iwaniec &
Kowalski [101], and Titchmarsh [200].
1.1. Motivation: prime numbers
A prime number is a positive integer n > 1 without proper divisors (in N).
The prime numbers are the multiplicative atoms of the integers: any positive
integer can be written as a unique product of powers of distinct primes (up
to the order of the factors). This fact is called the unique prime factorization
of the integers. Euclid (Prop. 20 in Elements 9; around 300 B.C.) proved
that there are infinitely many prime numbers as follows: if 2, p1 , . . . , pn are
prime numbers, then the number
Q := 2 · p1 · . . . · pn + 1
has a prime divisor q different from 2, p1 , . . . , pn (since otherwise q would
divide any linear combination of Q and q, in particular, +1).
An analytic version of the unique prime factorization is given by the identity
−1
Y
X 1
1
=
1− s
(1.1)
,
ns
p
p
n∈N
where the product is taken over all primes (a proof will be given later). Both,
the series and the product converge for s > 1 (also this will be proved below).
The identity between the series and the product was discovered by Euler [51]
in 1737. It gives a first glance on the intimate connection between the prime
numbers and certain objects in analysis. A first immediate consequence is
Euler’s proof of the infinitude of the primes. Assuming that there were only
1
Chapter 1
2
Classical L-functions
finitely many primes, the product in (1.1) is finite, and therefore convergent
throughout the whole complex plane, contradicting the fact that the series
reduces to the divergent harmonic series as s → 1+. Hence, there exist
infinitely many prime numbers. This argument might be slightly more complicated than Euclid’s elementary proof but, as we shall see later, the analytic
access yields much deeper knowledge on the distribution of the prime numbers. In fact, the series in (1.1) defines the famous Riemann zeta-function
which encodes many arithmetic information in its value distribution.
In view of the infinitude of the primes it is natural to ask how they are
distributed among the integers. It was the young Gauss who conjectured in
1791 (see Tagebuch, Werke, vol. 10.1) for the number π(x) of primes p ≤ x
the asymptotic formula
π(x) ∼ Li (x),
(1.2)
where
(1.3)
Li (x) :=
Z
0
x
du
:= lim
ǫ→0+
log u
Z
1−ǫ
0
+
Z
x
1+ǫ
du
log u
is the logarithmic integral. This would imply that, in first approximation,
the number of primes ≤ x is asymptotically logx x , and so the primes form a
set of zero density in N. It is recorded that Gauss came to his conjectural
asymptotic formula by calculating the number of primes up to several millions. However, there is also a heuristic argument in favor for his conjecture
by exploiting identity (1.1). For this aim we cut the product and the series
at x (assuming that this still leads to an asymptotic identity as x → ∞) and
let s = 1. This yields
−1
!
X
X 1 Y
1
1
∼
1−
= exp −
log 1 −
n
p
p
p≤x
p≤x
n≤x
!
X1
+ O(1) .
= exp
p
p≤x
By the well-known asymptotics for the truncated harmonic series,
X1
1
(1.4)
,
= log x + C + O
n
x
n≤x
where C := limN →∞ {
constant, we get
(1.5)
1
n≤N n
P
− log N} = 0.577 . . . is the Euler-Mascheroni
X1
p≤x
p
∼ log log x.
Section 1.1
Prime numbers
3
This formula is indeed true and was first obtained by Euler [51] in the form
1 1 1
+ + + . . . = log log ∞;
2 3 5
however, his proof had some gaps and the first waterproof argument is due to
Mertens [145]. Certainly, this asymptotic formula cannot be deduced from
Euclid’s proof. In particular, it shows that the sum over the reciprocals of the
prime numbers diverges, indicating that there are quite many primes (more
P
than squares since n 1/n2 < ∞). Using the Stieltjes integral (resp. partial
summation, a technique we meet later in detail), we also find
Z x
X1 Z x 1
π(u)
=
dπ(u) ∼
du.
p
u2
2 u
2
p≤x
Inserting Gauss’ conjectural asymptotics (1.2) shows that this is indeed of
the same size as predicted by Euler’s formula (1.5). Clearly, this is not a
proof but it might suggest that (1.2) indicates the correct order for the prime
counting function π(x).
Further evidence was found by Chebyshev [32, 33] around 1850 who proved
by elementary means that for sufficiently large x
0.921 . . . ≤ π(x)
log x
≤ 1.055 . . . .
x
Moreover, he showed that if the limit
log x
x→∞
x
exists, the limit is equal to one, which supports relation (1.2). For a proof of
these results and also for more details on the history of the theory of prime
number distribution we refer to Narkiewicz [159].
lim π(x)
There exist plenty of problems concerning prime numbers which are easy
to formulate but rather difficult to solve. Here is a short list of four famous
problems concerning the distribution of prime numbers.
• Does there exist an exact formula for the number π(x) of primes
p ≤ x? Is there an explicit formula for the nth prime number?
• Given a positive integer B ≥ 2, are there infinitely many pairs of
consecutive prime numbers having a difference ≤ B? (For B = 2
this is the famous twin prime conjecture!)
• Can any positive number be written as the sum of three primes?
Can any even integer greater than 2 be written as the sum of two
primes? (The second question is the open Goldbach conjecture!)
• Is there always a prime number in between two squares of positive
integers? (Having a view on the first primes we might expect a
positive answer.)
4
Chapter 1
Classical L-functions
We shall discuss the state of art of these problems later in these notes; we
may regard them as indicator what can be done and what cannot be done
with present day methods.
Another natural question is how the prime numbers are distributed in
residue classes (of course, this makes only sense for classes a mod m with
coprime a, m). One may try to mimic Euclid’s proof of the infinitude of
primes and, indeed, one can show that there are infinitely many prime numbers p ≡ 1 mod 4; however, one cannot succeed in proving the same for the
residue class 3 mod 5. M.R. Murty [153] gave a characterization of all prime
residue classes a mod m for which a Euclid-type proof exists; he showed that
a necessary and sufficient condition is that a2 ≡ 1 mod m.
In 1837, Dirichlet proved that there are infinitely many primes in any prime
residue class. His ingenious argument relies on a family of identities similar
to (1.1) and analytic properties of the appearing series, named Dirichlet Lfunctions. His approach is regarded as the beginning of analytic number
theory and it also marks the beginning of the theory of L-functions; it is
legend that the capital ‘L’ in the word ‘L-function’ stands for one of his
initials (Peter Gustav Lejeune Dirichlet).
For short, the idea of analytic number theory can be described as follows:
given an arithmetic function,
f : N→C,
n 7→ f (n),
one hopes to get arithmetic information about f by studying the analytic
behaviour of the generating function
Lf (s) :=
∞
X
f (n)
n=1
ns
;
in honour of Dirichlet’s contribution the generating series are called Dirichlet
series. It turns out that this is a rather fruitful concept. The set A of
arithmetic functions forms a commutative ring with respect to the standard
addition (f + g)(n) := f (n) + g(n) and the convolution (multiplication)
X
(f ∗ g)(n) :=
f (d)g(n/d),
d|n
where, as usual, we write d | n if the integer d divides the integer n, and d ∤ n
otherwise. These operations correspond to the addition and the multiplication of the associated Dirichlet series:
Lf +g (s) = Lf (s) + Lg (s)
and
Lf ∗g (s) = Lf (s) · Lg (s).
The set D of associated Dirichlet series forms a ring isomorphic to A, and
convolution identities in arithmetic (which play a centrale role in elementary
Section 1.1
Prime numbers
5
number theory) correspond one-to-one to product identities of Dirichlet series. This leads via (formal) differentiation to new identities for arithmetic
functions from old ones. Furthermore, in many cases one can exhibit number
theoretical information from identities for the associated Dirichlet series and
their analytic behaviour.
In number theory we are often concerned with multiplicative arithmetic
functions; their associated Dirichlet series can be written as an infinite product over the prime numbers and this is the essential property of an L-function.
In the next section we will present the prototype.
Exercise 1. Let x ≥ 3. Prove that there are more than (log 2)−1 log log x many
prime numbers p ≤ x.
Hint: use Euclid’s proof and induction.
Exercise 2. Prove that there are intervals of arbitrary length in (0, +∞) free of
prime numbers.
Exercise 3. Show that, for x > 1,
Z x
1
du
Li (x) =
1−
+ log log x + C
u log u
1
N
k!
x
x X
+O
,
=
log x
(log x)k
(log x)N +2
k=0
where C is a constant; in fact, one can show that C is the Euler-Mascheroni
constant, can you?.
Exercise 4. Prove
X1
p≤x
p
≥ log log x + O(1);
this is half way to the asymptotic formula (1.5).
Hint: Start to show the inequalities
X
Z x
Y
1
du
1
1−
>
.
≥
p
n
u
1
p≤x
n≤x
Euler’s ϕ-function ϕ(n) counts the number of prime residue classes mod n, i.e.,
ϕ(n) = ♯{1 ≤ a ≤ n : gcd(a, n) = 1}.
Exercise 5. i) Show that ϕ(n) = n − 1 if and only if n is prime.
ii) Prove that
Y
1
.
ϕ(n) = n
1−
p
p|n
Hint: consider first n = pk with p prime.
6
Chapter 1
Classical L-functions
Exercise 6. Let q > 1 and x = qk + r with 0 ≤ r < q be positive integers. Prove
that
x
ϕ(q)
π(x) ≤ q + kϕ(q) + r ≤ 2q +
q
and
π(x)
ϕ(q)
lim sup
≤
.
x
q
x→∞
Deduce that π(x) = o(x). (This argument is due to Fousserau, cf. [159]).
The sieve of Eratosthenes is a very efficient algorithm to produce a list of all
prime numbers below a given magnitude:
Exercise 7. * Make a list of all positive integers 2 ≤ n ≤ x and mark all proper
√
multiples of prime numbers p ≤ x. Then the number of unmarked numbers is
π(x). Why?
ii) Prove
Y
1
+ O(2y )
(1.6)
π(x) − π(y) ≤ x
1−
p
p≤y
√
for any 1 ≤ y ≤ x (maybe with help of some literature, e.g., [159]).
iii) Use ii) to show that, for sufficiently large x,
x
.
π(x) ≪
log log x
Hint: recall Exercise 4.
Exercise 8. i) Prove that there are infinitely many primes p ≡ 1 mod 4 and 3
mod 4 (one case is rather tricky and involves the theory of quadratic residues).
Hint: For one case one may use a fact from the theory of quadratic residues:
the congruence X 2 ≡ −1 mod p with a prime p 6= 2 is solvable if and only if
p ≡ 1 mod 4.
ii) What can be done for the prime residue classes mod 6 and mod 10?
The Möbius µ-function is defined by setting µ(1) = 1, µ(n) = (−1)ν if n is the
product of ν distinct primes, and µ(n) = 0 otherwise, i.e., if n has a quadratic
prime divisor.
Exercise 9. i) Show that
X
µ(d) =
d|n
1
0
if n = 1,
otherwise,
ii) Prove the Möbius inversion formula: for two arithmetic functions f and g, the
statement
X
f (n) =
g(d)
d|n
is equivalent to
g(n) =
X
d|n
µ(d)f (n/d).
Section 1.2
Riemann’s zeta-function
7
Exercise 10. Prove all claims about the commutative ring A of arithmetic functions (for the commutativity one needs Möbius inversion formula). What is the
neutral element in this ring with respect to convolution? Prove the isomorphy between the ring of arithmetic functions A and the ring D of associated Dirichlet
series! Finally, give a characterization of the invertible elements in these rings!
Hint: for some help see [2].
1.2. Riemann’s zeta-function
√
Let s = σ + it with σ, t ∈ R and i := −1 be a complex variable (this
mixture of greek and latin letters have become tradition since their use in
Landau’s papers). The Riemann zeta-function is given by
(1.7)
∞
X
1
ζ(s) =
.
ns
n=1
This series was studied ever since the fundamentals of calculus were laid.
One of the most famous question in the early 18th century was about the
value of ζ(2) found by Euler in 1737. Euler considered only real s in his
studies but Riemann was the first to investigate the Riemann zeta-function
as a function of a complex variable. In his only one but outstanding paper
[175] on number theory from 1859, Riemann outlined how Gauss’ conjecture
(1.2) could be proved by using the function ζ(s). As a matter of fact, it is
the complex-analytic point of view that allows to get deeper knowledge about
the zeta-function (and which therefore was unattainable for Euler). However,
at Riemann’s time the theory of functions was not developed so far, but the
open questions concerning the zeta-function pushed the research in this field
quickly forward.
1.2.1. The half-plane of absolute convergence. It is easily seen (by
Riemann’s integral test) that the series (1.7) defining zeta converges absolutely for σ > 1. Since, for σ ≥ σ0 > 1,
∞
∞ Z n
∞
X
X
X
1
du
1
≤
1
+
≤
s
σ
n n0
uσ0
n=1
n=2 n−1
n=1
Z ∞
1
,
= 1+
u−σ0 du = 1 +
σ0 − 1
1
the series in question converges uniformly in any compact subset of the halfplane of absolute convergence σ > 1. A well-known theorem of Weierstrass
states that the limit of a uniformly convergent sequence of analytic functions
is analytic (see Titchmarsh [199], §2.8). Hence, ζ(s) is analytic for σ > 1.
This reasoning holds far more general for Dirichlet series: in general, Dirichlet
8
Chapter 1
Classical L-functions
series converge in half-planes (provided that they do converge) and define
analytic functions in their half-plane of uniform convergence.
Recall from the introduction identity (1.1) linking the prime numbers and
the zeta-function. The product over the primes is called Euler product in
honour of its discoverer. Our next aim is to verify this fundamental Euler
product representation. Let σ > 1. In view of the unique prime factorization
of the integers and the geometric series expansion,
−1 Y X 1
Y
1
1
1
.
=
1 + s + 2s + . . . =
1− s
p
p
p
ns
n
p≤x
p≤x
p|n⇒p≤x
Since
∞
Z ∞
X 1 X 1
X 1
x1−σ
du
<
−
≤
=
s
ns n>x nσ
uσ
σ−1
x
n
n=1 n
p|n⇒p≤x
tends to zero as x → ∞, we obtain identity (1.1) by sending x → ∞. Summing up, we have just proved
Theorem 1.1. ζ(s) is analytic for σ > 1 and satisfies in this half-plane the
identity
−1
∞
Y
X
1
1
=
1− s
.
(1.8)
ζ(s) =
s
n
p
p
n=1
Later we shall see more identities between Dirichlet series and Euler products,
each of them will allow us to study a certain arithmetic object (encoded in
the Euler product) by means of analysis (via Dirichlet series).
1.2.2. Riemann’s memoir - proven facts. Now we study Riemann’s
famous memoir [175]. Actually, he proved only two statements. First of all,
Riemann showed that the function
1
ζ(s) −
s−1
is entire; thus, ζ(s) has an analytic continuation throughout the whole complex plane except for a simple pole at s = 1 with residue 1 (corresponding
to the divergent harmonic series). Secondly, Riemann proved the functional
equation for the zeta-function: for all s ∈ C,
s
1−s
− 1−s
− 2s
ζ(s) = π 2 Γ
ζ(1 − s).
(1.9)
π Γ
2
2
This shows a point symmetry for the function defined by the left-hand side
with respect to the point s = 21 . In view of the Euler product (1.8) it is easily
seen that ζ(s) has no zeros in the half-plane σ > 1. Using the functional
Section 1.2
Riemann’s zeta-function
9
equation (1.9), it turns out that ζ(s) vanishes in σ < 0 exactly at the socalled trivial zeros
ζ(−2n) = 0
for
n ∈ N,
all of them being simple. This follows from some basic properties of
the Gamma-function. By Gauss’ product representation for the Gammafunction,
(1.10)
N!N z
,
N →∞ z(z + 1)(z + 2) · . . . · (z + N)
Γ(z) = lim
Γ(z) has simple poles for z = 0, −1, −2, . . . and no zeros at all. In order to
compensate the poles of Γ( 2s ) in (1.9) for s = −2n, ζ(s) has to vanish there.
The behaviour of ζ(s) is quite well understood in all of the complex plane
but the so-called critical strip 0 ≤ σ ≤ 1 (which justifies to call this strip
critical).
0.075
0.05
0.025
-14
-12
-10
-8
-6
-4
-2
-0.025
-0.05
-0.075
-0.1
Figure 1. The graph of ζ(s) for s ∈ [−14.5, 0].
All other zeros of ζ(s) are said to be nontrivial, and it comes out that they
are all non-real (and that there location is in fact a nontrivial task). We
denote the nontrivial zeros by ρ = β + iγ. Obviously, they have to lie in
the critical strip 0 ≤ σ ≤ 1. The functional equation, in addition with the
identity
ζ(s) = ζ(s),
show some symmetries of ζ(s). In particular, the nontrivial zeros of ζ(s)
have to be distributed symmetrically with respect to the real axis and the socalled critical line σ = 12 . It was Riemann’s ingenious contribution to number
theory to point out how the distribution of these nontrivial zeros is linked to
the distribution of prime numbers.
1.2.3. Analytic continuation. To set the stage for the further discussion of Riemann’s memoir, we shall sketch a proof of his first result concerning
the meromorphic continuation of ζ(s). At s = 1 the series defining the zetafunction reduces to the harmonic series. For an analytic continuation for ζ(s)
we have to seperate this singularity. For this purpose we shall make use of
Chapter 1
10
Classical L-functions
Lemma 1.2. Let λ1 < λ2 < . . . be a divergent sequence of real numbers,
P
define for αn ∈ C the function A(u) := λn ≤u αn , and let F : [λ1 , ∞) → C
be a continuous differentiable function. Then
Z x
X
αn F (λn ) = A(x)F (x) −
A(u)F ′(u) du.
λ1
λn ≤x
This switch from a sum to an integral is called Abel’s partial summation. It
is an important technical tool in analytic number theory: often integrals are
easier to handle than sums. The reader who is familiar with the RiemannStieltjes integral may skip the proof.
Proof. We have
A(x)F (x) −
X
αn F (λn ) =
λn ≤x
X
λn ≤x
=
αn (F (x) − F (λn ))
XZ
λn ≤x
x
αn F ′ (u) du.
λn
Since λ1 ≤ λn ≤ u ≤ x, interchanging integration and summation yields the
assertion. •
Now we apply partial summation to finite pieces of the Dirichlet series
defining zeta. Let N < M be positive integers and σ > 1. Then, applying
Lemma 1.2 with F (u) = u−s , αn = 1 and λn = n, yields
Z M
X 1
[u]
1−s
1−s
= M
−N
+s
du
s
s+1
n
N u
N <n≤M
Z M
Z M
[u] − u
du
1−s
1−s
= M
−N
+s
du + s
s+1
u
us
N
N
Z M
1
[u] − u
=
(N 1−s − M 1−s ) + s
du;
s−1
us+1
N
here, as usual, we write [u] for the largest positive integer less than or equal
to u. Sending M → ∞, we obtain
Z ∞
X 1
N 1−s
[u] − u
(1.11)
ζ(s) =
+
+s
du.
s
s+1
n
s
−
1
u
N
n≤N
Since −1 < [u] − u ≤ 0, it follows that the integral exists for any s with σ > 0
(and any value for N). Thus we have proved
Theorem 1.3. For σ > 0,
s
+s
ζ(s) =
s−1
Z
1
∞
[u] − u
du.
us+1
Section 1.2
Riemann’s zeta-function
11
Hence, ζ(s) has an analytic continuation to the half-plane σ > 0 except for a
simple pole at s = 1 with residue 1.
By the functional equation (1.9) we obtain a meromorphic continuation for
the zeta-function to the whole complex plane (however, we postpone the proof
of the functional equation to Chapter 2). Taking into account properties of
the Gamma-function it turns out that the only singularity of ζ(s) is the simple
pole at s = 1. This proves Riemann’s first statement (subject to the validity
of (1.9)).
1.2.4. Riemann’s memoir - the conjectures. More spectacular than
Riemann’s proven results are his conjectures. First of all, for the number
N(T ) of nontrivial zeros ρ = β + iγ with 0 < γ ≤ T (counted according
multiplicities) he conjectured the asymptotic formula
N(T ) ∼
T
T
log
;
2π
2πe
this was proved in 1895/1905 by von Mangoldt [141, 142] who found more
precisely
(1.12)
N(T ) =
T
T
log
+ O(log T ).
2π
2πe
Hence there are infinitely many nontrivial zeros and their frequency increases
with their imaginary parts. Riemann’s second conjecture was about the horizontal distribution of the nontrivial zeros. Riemann worked with ζ( 21 + it)
and wrote
”...und es ist sehr wahrscheinlich, dass alle Wurzeln reell sind.
Hiervon wäre allerdings ein strenger Beweis zu wünschen; ich
habe indess die Aufsuchung desselben nach einigen flüchtigen
vergeblichen Versuchen vorläufig bei Seite gelassen...”
which means that very likely all roots t are real, i.e., all nontrivial zeros lie on
the so-called critical line σ = 21 . This is the famous, yet unproved Riemann
hypothesis. It had been in Hilbert’s famous list of 23 problems for the 20th
century and it is now one of the seven millennium problems. It should be
noticed that Riemann also calculated the first three zeros (i.e., with respect
to their imaginary parts in the upper half-plane, ordered by their size); the
first one is ρ = 12 + i · 14.134 . . ..
Further, Riemann conjectured that there exist some constants A, B such
that
s
Y
s
s
1
− 2s
ζ(s) = exp(A + Bs)
1−
exp
.
s(s − 1)π Γ
(1.13)
2
2
ρ
ρ
ρ
Chapter 1
12
Classical L-functions
1.5
1
0.5
-1
1
2
3
-0.5
-1
-1.5
Figure 2. The values of ζ(1/2 + it) for 0 ≤ t ≤ 50.
His final conjecture relates the prime numbers with the zeros of the zetafunction. The so-called explicit formula states that
(1.14)
π(x) +
1
∞
X
π(x n )
n=2
n
= Li(x) −
+
Z
x
X
Li(xρ ) + Li(x1−ρ )
ρ=β+iγ
γ>0
∞
u(u2
du
− log 2
− 1) log u
1
for any x ≥ 2 not being a prime power (otherwise a term 2k
has to be added
k
on the left hand-side, where x = p ); the appearing logarithmic integral has
to be defined carefully by analytic continuation from (1.3). This was proved
in 1895 by von Mangoldt [141] whereas the last but one conjecture was
proved by Hadamard [72]. The explicit formula follows from both product
representations of ζ(s), the Euler product on one side and the Hadamard
product over the zeros on the other side.
Riemann’s ideas led to the first proof of Gauss’ conjecture (1.2), the celebrated prime number theorem, by Hadamard [73] and de La Vallée Poussin
[202] (independendly) in 1896. Later in this chapter we will prove the prime
number theorem and all of Riemann’s conjectures (the Hadamard product
representation and the explicit formula in this chapter, the functional equation in the following chapter, and the Riemann-von Mangoldt formula in
Chapter 3) – except his “hypothesis”. However, first we travel back in time
and study Dirichlet’s approach to the problem of prime number distribution
in arithmetic progressions.
Exercise 11. Deduce from the prime number theorem in the form π(x) ∼ x/ log x
that
X log p
X1
= log x + O(1)
and
= log log x + O(1).
p
p
p≤x
Hint: partial summation.
p≤x
Section 1.2
Riemann’s zeta-function
13
Exercise 12. The following evaluation of ζ(2) by elementary means is due to
Calabi: verify
∞
∞ Z 1Z 1
∞
X
X
1
3X 1
=
=
x2m y 2m dx dy
4
n2
(2m + 1)2
0
m=0
m=0 0
n=1
Z 1Z 1 X
Z 1Z 1
∞
dx dy
2m
=
(xy) dx dy =
.
2 2
0
0
0
0 1−x y
m=0
Use the transformation
sin v
sin u
and y =
cos v
cos u
in order to compute the appearing double integral above and deduce
x=
ζ(2) =
∞
X
π2
1
=
.
n2
6
n=1
Exercise 13. * i) This provides an alternative analytic continuation for the zetafunction: Prove
∞
X
(−1)n−1
1
ζ(s) =
1 − 21−s n=1
ns
(1.15)
and show that the alternating series on the right-hand side converges for σ > 0.
Thus, in view of the functional equation (1.9), this yields a meromorphic continuation to the whole complex plane. Where are possible singularities? None in the
half-plane of convergence but a simple pole at s = 1 of residue 1 since 1 − 21−s
vanishes for s = 1 and 1 − 12 + 13 ∓ . . . = log 2; however, the other zeros of 1 − 21−s
do not lead to singularities – why?.
Hint: consider the series


 X
X  1
−2
.

 ns
n6=0 mod 3
n≡0 mod 3
ii) Use (1.15) to show that ζ(s) < 0 for 0 ≤ s < 1.
Exercise 14. Show that |ζ(s)| ≤ 2|s| for σ ≥ 21 .
Exercise 15. Show that the multiplicity of any nontrivial zero ρ = β + iγ is
bounded above by log |γ|.
Hint: use the Riemann-von Mangoldt formula (1.12).
Exercise 16. Find representations in terms of the zeta-function for
(1.16)
Lϕ (s) =
∞
X
ϕ(n)
n=1
ns
and
where ϕ is Euler’s ϕ-function and τ (n) :=
Lτ (s) =
∞
X
τ (n)
n=1
P
d|n 1
ns
,
is the divisor function.
14
Chapter 1
Classical L-functions
1.3. Dirichlet L-functions
A special role in number theory is played by multiplicative arithmetic functions and their associated generating series. Multiplicative functions respect
the multiplicative structure of N: an arithmetic function f is called multiplicative if f (1) 6= 0 and
f (m · n) = f (m) · f (n)
for all coprime integers m, n; if the latter identity holds for all integers, f
is said to be completely multiplicative. The generating Dirichlet series associated with a completely multiplicative function has, at least in a formal
way, an Euler product representation similar to the one for the Riemann
zeta-function. However, in this section we shall specify to a concrete family
of completely multiplicative functions introduced by Dirichlet [47] in 1837
in order to prove that there are infinitely many primes in any prime residue
class.
1.3.1. Characters. A character χ is a non-trivial group homomorphism
from a finite (for the sake of simplicity) abelian group G onto C∗ . By the
structure theorem for finite abelian groups any such group G is the direct
product of cyclic groups. Later we will be concerned with the the multiplicative group of the ring of residue classes mod q, i.e., the group of prime
residue classes modulo q,
(Z/qZ)∗ := {a mod q : gcd(a, q) = 1}.
By the chinese remainder theorem,
(Z/qZ)∗ =
Y
(Z/pν(q;p) Z)∗ ,
p|q
where ν(q; p) denotes the exponent of the prime p in the prime factorization of
the integer q. In this case the decomposition into a product of cyclic groups is
much easier to obtain. Gauss proved that the group of residue classes modulo
q is cyclic if and only if q = 2, 4, pν or 2pν , where p 6= 2; a generator of such
a cyclic group (Z/qZ)∗ is called a primitive root mod q. In the case q = 2ν
one has
(Z/2ν Z)∗ = h−1i × h5i
(which leads to a cyclic group if ν = 1, 2, since then −1 ≡ 5 mod 22 ). In any
case, the group of prime residue classes mod q is a product of finitely many
cyclic groups.
For the first we shall argue more generally. Assume that
G=
r
Y
j=1
Gj
with Gj = hgj i.
Section 1.3
Dirichlet L-functions
15
In particular, any g ∈ G has a unique representation of the form
g=
r
Y
t
gj j
0 < tj ≤ ℓj ,
with
j=1
where ℓj = ♯Gj is the group order of Gj . Since a character on G is a group
homomorphism, i.e.,
χ(a · b) = χ(a) · χ(b)
for all a, b ∈ G,
it follows that
χ(g) =
r
Y
χ(gj )
tj
for
g=
j=1
r
Y
t
gj j .
j=1
Therefore, a character is uniquely determined by its values on the generators.
By a theorem of Lagrange, the order of any element of a finite abelian group
is a divisor of the group order (in the particular case of the group of prime
residue classes this is an older theorem of Fermat and Euler). Hence,
ℓ
1 = χ(1) = χ(gj j ) = χ(gj )ℓj ,
and thus χ(gj ) is an ℓj -th root of unity, i.e.,
kj
for some kj ∈ Z with 0 < kj ≤ ℓj .
χ(gj ) = exp 2πi
ℓj
Consequently, there are at most ℓ1 · . . . · ℓr many characters χ on G. On
the contrary, any choice of k1 , . . . , kr with 0 < kj ≤ ℓj defines via χ(gj ) =
k
exp(2πi ℓjj ) such a character. Hence, the number of characters χ on G is equal
to the group order ♯G = ℓ1 · . . . · ℓr .
We may define the product of two characters mod q by setting
(χ · ψ)(g) = χ(g) · ψ(g);
this gives the set of characters χ mod q the structure of a group, the character
group (resp. dual group) of G, for short Ĝ. Its unit element, the principal
character, is the character constant 1 and is denoted by χ0 . Since |χ(g)| = 1,
the inverse of a character χ ∈ Ĝ is given by
χ(g) = χ(g) = χ(g)−1.
Given
χk (gj ) =
(
exp 2πi ℓ1j
1
if j = k,
otherwise,
the mapping gj 7→ χj is an isomorphism between G and its character group
Ĝ. We illustrate these observations with the example G = (Z/5Z)∗ :
Chapter 1
16
0
1≡2
2 ≡ 21
4 ≡ 22
3 ≡ 23
Classical L-functions
χ0 χ1 χ2 χ3
+1 +1 +1 +1
+1 -1 +i -i
+1 +1 -1 -1
+1 -1 -i +i
We find h2i ∼
= hχ2 i (of course, here we can also replace 2 by 3 or χ2 by χ3 ).
1.3.2. The orthogonality relations. Next we shall prove the important orthogonality relations for characters, the heart of Dirichlet’s method.
Lemma 1.4. For g ∈ G,
1 X
χ(g) =
1
0
if g = 1,
otherwise,
1 X
χ(g) =
♯G g∈G
1
0
if χ = χ0 ,
otherwise.
♯G
and, for χ ∈ Ĝ,
χ∈Ĝ
Proof. Given χ 6= χ0 , there exists an element h ∈ G with χ(h) 6= 1. Since
with g also gh runs through G, we get
X
X
X
χ(g).
χ(gh) = χ(h)
χ(g) =
g∈G
g∈G
g∈G
P
Hence,
g∈G χ(g) = 0. The case χ = χ0 is trivial. The second formula
follows in a similar way or, alternatively, via the isomorphism G ∼
= Ĝ. •
Using Lemma 1.4 with g −1 a in place of g resp. with ψχ instead of χ and
noting that χ(g −1 a) = χ(g)χ(a), we obtain
Lemma 1.5. For g, a ∈ G,
1 X
1
χ(g)χ(a) =
0
♯G
χ∈Ĝ
and, for χ, ψ ∈ Ĝ,
1 X
ψ(g)χ(g) =
♯G g∈G
1
0
if g = a,
otherwise,
if χ = ψ,
otherwise.
Now we restrict to groups of prime residue classes (Z/qZ)∗ . Via the natural
embedding of (Z/qZ)∗ in Z we can define characters χ mod q on the whole
of Z by setting
χ(n + qZ)
if gcd(n, q) = 1,
χ(n) =
0
otherwise.
Section 1.3
Dirichlet L-functions
17
The new objects are called Dirichlet characters χ mod q. The function n 7→
χ(n) is completely multiplicative; moreover, it is a q-periodic function on Z,
i.e., χ(n + q) = χ(n) for any n ∈ Z. Notice that ♯(Z/qZ)∗ = ϕ(q). The
orthogonality relation for characters takes therefore the form: if a and q are
coprime, then
X
1
1
if n ≡ a mod q,
χ(a)χ(n) =
(1.17)
0
otherwise.
ϕ(q)
χ mod q
With this tool we can sieve prime residue classes from the set of positive
integers. In view of the divergence of the sum of the reciprocals of the primes
we shall investigate the formal identity
(1.18)
X
p≡a mod
X
X χ(p)
1
1
=
.
χ(a)
p
ϕ(q)
p
p
χ mod q
q
If we can prove the divergence of the expression on the right-hand side, then
there are infinitely many prime numbers p ≡ a mod q. Of course, this makes
only sense if we assume a and q to be coprime.
1.3.3. Dirichlet’s prime number theorem for arithmetic progressions. For σ > 1, the Dirichlet L-function L(s, χ) associated with a character
χ mod q is given by
−1
∞
X
χ(p)
χ(n) Y
=
1− s
;
L(s, χ) =
ns
p
p
n=1
the proof of the identity between the Dirichlet series and the Euler product
follows along the lines of Theorem 1.1. In the special case of the principal
character χ0 mod q we obtain
−1
Y
Y
1
1
(1.19)
L(s, χ0 ) =
1− s
= ζ(s)
1− s ;
p
p
p∤q
p|q
in particular, we may regard ζ(s) as the Dirichlet L-function to the principal
character χ0 mod 1., and also for larger moduli q the Dirichlet L-function to
principal characters have a similar analytic behaviour as the zeta-function.
Theorem 1.6. Let χ mod q be a character 6= χ0 . Then, the series
P∞
−s
converges in σ > 0 and uniformly in any compact subset;
n=1 χ(n)n
in particular, L(s, χ) is analytic in σ > 0.
Notice that the series defining L(s, χ) cannot converge absolutely in σ ≤ 1
(and hence the Euler product representation for L(s, χ) is not valid inside
the critical strip).
Chapter 1
18
N <n≤M
P
χ(n) ≪ 1. Partial summation shows
Z M
A(M) A(N)
|s|
χ(n)
A(u)
N −σ .
=
−
+s
du ≪ 1 +
s+1
ns
Ms
Ns
u
σ
N
Proof. Clearly, A(x) :=
X
Classical L-functions
n≤x
This implies the convergence; the other assertions of the theorem follow as
in the case of the zeta-function. •
In particular, L(s, χ) is regular in s = 1 if and only if χ 6= χ0 . In view
of the Euler product representation there are no zeros of L(s, χ) in s >
1. Consequently, we can define the logarithm of Dirichlet L-functions (by
choosing any of its branches). We find, for σ > 1,
(1.20)
log L(s, χ) =
∞
XX
χ(p)k
p
k=1
kpks
=
X χ(p)
ps
p
+ O(1).
In view of (1.18) we shall show that Dirichlet L-functions L(s, χ) do not
vanish at s = 1.
Theorem 1.7. For any character χ, we have L(1, χ) 6= 0.
This statement is the difficult part in Dirichlet’s argument [47]; however,
here we shall not give his original innovative but rather complicated proof for
which he developed the analytic class number formula, an identity relating
the value L(1, χ) as a finite sum with certain non-zero invariants on classes
of quadratic forms (for details of this approach we refer to Narkiewicz [159]).
We shall follow an argument of Mertens from 1897.
Proof. We may assume that χ is not the principal character. Let s > 1. It
follows from (1.20) and the orthogonality relation for characters (1.17) that
X
X
1
χ(a) log L(s, χ) =
ϕ(q) χ mod q
p
∞
X
k=1
pk ≡a mod q
1
≥ 0.
kpks
In particular, for a = 1,
(1.21)
Y
χ mod q
L(s, χ) ≥ 1.
Since L(s, χ0 ) has a simple pole at s = 1 (inherited from ζ(s) by (1.19)) and,
by Theorem 1.6, all other L(s, χ) are regular, it follows from (1.21) that there
is at most one character χ for which L(1, χ) = 0. Since
L(1, χ) = L(1, χ)
such a character has to be real, i.e., χ = χ.
Section 1.3
Dirichlet L-functions
19
P
Now suppose χ is real. Then we define f = χ ∗ 1, resp. f (n) = d|n χ(d)
(resp. Lf (s) = ζ(s)L(s, χ)). Obviously, f is multiplicative. We find f (pk ) = 1
if p divides q; otherwise, if p does not divide q, then

 k + 1 if χ(p) = +1,
k
f (p ) =
1
if χ(p) = −1 and k ≡ 0 mod 2,

0
if χ(p) = −1 and k ≡ 1 mod 2.
It follows that f (n) ≥ 0 and f (m2 ) ≥ 1. Therefore,
X 1
X f (m2 )
X f (n)
≥
,
≥
1
m
m
2
n
2
m≤N
m≤N
n≤N
which diverges, as N → ∞. On the contrary, partial summation implies
X χ(d) X 1
X 1 X χ(d)
X f (n)
=
1
1
1 +
1
1
2
n2
b 2 b≤N b 2
d2
d≤N d
N2
N2
n≤N 2
b≤
(1.22)
N <d≤
d
b
= 2NL(1, χ) + O(1).
Since the left-hand side diverges to +∞, this yields L(1, χ) 6= 0. This proves
the theorem. •
In order to prove the infinitude of primes in prime residue classes a mod q,
we introduce in (1.18) a variable s > 1. By (1.20), we have
X
X χ(p)
X 1
1
=
χ(a)
ps
ϕ(q)
ps
p
χ mod q
p≡a mod q
=
1
1 X
log L(s, χ0 ) +
χ(a) log L(s, χ) + O(1).
ϕ(q)
ϕ(q) χ6=χ
0
Sending s → 1+, the first term on the right-hand side diverges by (1.19),
and the second term converges with regard to Theorem 1.7. Hence, the series
on the left-hand side is divergent. Thus we have proved Dirichlet’s prime
number theorem for arithmetic progressions:
Theorem 1.8. Any prime residue class contains infinitely many prime numbers.
We resume: the divergence of the series over all reciprocals of primes p ≡
a mod q with coprime a and q was shown by exploiting the pole of L(s, χ0 )
at s = 1, so via (1.19) once more the pole of the zeta-function (as in Euler’s
proof of the infinitude of primes). As we shall see later on, much of the
machinery developed for the zeta-function in order to prove Gauss’ conjecture
(1.2), the celebrated prime number theorem, can (with slight modifications)
also be applied to Dirichlet L-functions. This will lead us to the following
generalization of the prime number theorem: let π(x; a mod q) denote the
20
Chapter 1
Classical L-functions
number of primes p ≤ x in the residue class a mod q; then, for a coprime
with q,
(1.23)
π(x; a mod q) ∼
1
π(x).
ϕ(q)
This shows that the primes are uniformly distributed in the prime residue
classes.
In 1853, Chebyshev claimed (in a letter to Fuss, cf. [67]) that there are,
in some sense, more primes in the residue class 3 mod 4 than in the class
1 mod 4, e.g., there are 4808 primes of the first type and only 4783 of the
second type below 100 000 and this bias seems to hold if we count more and
more primes. However, this claim is not true: Littlewood [136] showed that
there are arbitrarily large values of x such that
1
1 x2
log log x.
π(x; 1 mod 4) − π(x; 3 mod 4) ≥
2 log x
Nevertheless, assuming the generalized Riemann hypothesis (which will be
explained in the following paragraph), Rubinstein & Sarnak [176] proved
that Chebyshev’s claim holds for more than 99.59% of the values of x. In
general it is expected that such a phenomenon can be observed for any pair
of prime residue classes a, b mod q with a being a quadratic residue and b not
and that in the “prime number race” the primes p ≡ b mod q dominate over
those in a mod q. For a nice survey on this theme see Granville & Martin
[67].
1.3.4. Analytic theory of Dirichlet L-functions. Let χ be a character mod q. It is possible that for values of n coprime with q the character
χ(n) may have a period less than q. If so, we say that χ is imprimitive, and
otherwise primitive. If q is prime, then every character χ mod q is primitive.
If χ∗ is a primitive character mod q ∗ and q a multiple of q ∗ , then we can
construct via
∗
χ (n) if gcd(n, q) = 1,
χ(n) =
0
if gcd(n, q) > 1,
a character χ mod q, and χ is induced by χ∗ . We illustrate this by the
following example:
1 2 3
4 5
6 7 8
9 10
n mod 10
∗
χ (n) = +1 +i −i −1 0 +1 +i −i −1 0
χ(n) = +1 0 −i
0 0
0 +i 0 −1 0
Every imprimitive character is induced by a primitive one. Two characters
are non-equivalent if they are not induced by the same character. If χ∗ mod
Section 1.3
Dirichlet L-functions
21
q ∗ is a primitive character which induces another character χ mod q, then
Y
χ∗ (p)
∗
(1.24)
.
L(s, χ) = L(s, χ )
1−
ps
p|q
Being twists of the Riemann zeta-function with multiplicative characters,
Dirichlet L-functions share many properties with the zeta-function. For instance, there is an analytic continuation to the whole complex plane, only
with the difference that L(s, χ) is regular at s = 1 if and only if χ is nonprincipal (see Theorem 1.7). Furthermore, L-functions to primitive characters satisfy a functional equation of the Riemann-type; namely,
(1.25)
q s+δ
s+δ
1+δ−s
τ (χ) q 1+δ−s
2
2
Γ
Γ
L(s, χ) = δ √
L(1 − s, χ),
π
2
i q π
2
where δ := 21 (1 − χ(−1)) and
(1.26)
τ (χ) :=
X
χ(a) exp
a mod q
2πia
q
is the Gaussian sum attached to χ. One finds a setting for the zeros which is
quite similar to the one for zeta: the trivial zeros are those which correspond
to poles of the Gamma-factors in the functional equation; all other zeros
are said to be nontrivial and they lie in the critical strip. Also for Dirichlet L-functions it is expected that the analogue of the Riemann hypothesis
holds; more precisely: all nontrivial zeros of a Dirichlet L-function L(s, χ)
to a primitive character are conjectured to lie on the critical line. The restriction to primitive characters is made to exclude the zeros of the factor
Q
∗
−s
p|q (1 − χ (p)p ) in (1.24), which all lie on the line σ = 0.
Exercise 17. i) Let f be a multiplicative arithmetic function. Prove the formal
identity
∞
∞
X
f (n) Y X f (pk )
=
.
ns
pks
p
n=1
k=0
Moreover, if f is completely multiplicative, then
∞
X
f (n) Y
f (p) −1
.
=
1
−
s
ns
p
p
n=1
ii) Assume that f (n) ≪ nc for some non-negative constant c. Show that F (s) :=
P∞
−s converges in some half-plane σ > σ and defines there an analytic
a
n=1 f (n)n
function; find an explicit value for the abscissa of convergence.
Exercise 18. Prove that µ(n), τ (n), and ϕ(n) are multiplicative functions. Are
they also completely multiplicative? Can you prove Euler product representations
Chapter 1
22
Classical L-functions
for the associated Dirichlet series, i.e., for the functions in (1.16) as well as for
P∞
−s
n=1 µ(n)n ?
Exercise 19. For an odd prime

 +1
a
=
0

p
−1
p, the Legendre symbol modulo p is defined by
if X 2 ≡ a mod p
if p | a,
otherwise.
is solvable,
Prove that the Legendre symbol is a character mod p.
Hint: the squares in (Z/pZ)∗ form a subgroup of index 2.
Exercise 20. Determine all characters mod q for q = 10, 12, 16. Compute the
structure of the corresponding character groups.
The mean-value of arithmetic functions can often be computed by counting lattice point subject to some side-conditions. One of the basic techniques is Dirichlet’s
hyperbola method.
P
Exercise 21. * i) For the divisor function τ (n) = d|n 1 show that
1
X
τ (n) = x log x + x(2C − 1) + O x 2 ,
n≤x
where C is the Euler-Mascheroni constant.
Hint: note that the left hand side counts the number of integral lattice points under
a hyperbola and write for this
X
X
X
X
1.
1=
1+
1−
bd≤x
bd≤x
√
d≤ x
bd≤x
√
b≤ x
√
b,d≤ x
ii) Verify all steps in identity (1.22).
Exercise 22. * Let a and q be coprime. Prove that
X
1
1
1
∼
.
s
p
ϕ(q) s − 1
p≡a mod q
and
X
ϕ(q)
p≤x
p≡a mod
q
1
− log log x ≪ 1.
p
Can you use the latter estimate to find an upper bound for the least prime p ≡
a mod q?
Exercise 23. * Let χ be the non-principal character modulo 4. Observe that the
factors in the Euler product
Y
χ(p)
1−
p
p6=2
are greater than 1 for primes p ≡ 3 mod 4 and less than 1 for p ≡ 1 mod 4. What
is the value of this product? How can this value be used as support for Chebyshev’s
claim on the existence of more primes p ≡ 3 mod 4 than p ≡ 1mod; 4?
Section 1.3
Dirichlet L-functions
23
As a matter of fact, Euler already had an analytic “proof” for the infinitude of
primes in the prime residue classes mod 4 (see Weil [211]). His argument shall be
recovered in the following
Exercise 24. * Let χ denote the non-principal character mod 4. Prove that
Y p + χ(p)
.
2=
p − χ(p)
p
Deduce that
X (−1)χ(p) 1 X (−1)χ(p)
1
log 2 =
+
+ ....
2
p
3 p
p3
p
Use Maple or Mathematica in order to find that
X (−1)χ(p)
= 0.33498 . . . ;
p
p
deduce that there are infinitely many primes in any prime residue class mod 4.
Exercise 25. * i) Let χ be a character modulo q and denote by τ (χ) the associated
Gauss sum. Show that, for n and q coprime,
X
an
χ(a) exp
χ(n)τ (χ) =
;
q
a mod q
if χ is primitive, then this identity holds for all n.
ii) For a primitive character χ mod q, prove that |τ (χ)|2 = q.
Hint: use i) (or search for help in [2]).
The Polya-Vinogradov inequality states that characters cannot be constant on a
long sequence of consecutive integers:
Exercise 26. * Let χ be a non-principal character modulo q. Prove that
X
1
χ(n) ≤ 2q 2 log q.
n≤N
Hint: use the previous exercise to substitute the appearing character by trigonometric expressions.
A function has at most one Dirichlet series representation:
Exercise 27. * i) Assume that
A(s) =
∞
X
a(n)
n=1
ns
and
B(s) =
∞
X
b(n)
n=1
ns
are two Dirichlet series converging in some half-plane σ > σa . Prove that if there
is a region in this half-plane for which A(s) = B(s), then a(n) = b(n) for all n.
ii) Deduce from i) that any convergent Dirichlet series has a zero-free half-plane.
Chapter 1
24
Classical L-functions
1.4. The prime number theorem
It was Riemann’s contribution which led to the proof of Gauss’ conjecture
(1.2), the prime number theorem. After substantial work by von Mangoldt
and others Hadamard [73] and de la Vallée-Poussin [202] gave the first proof
(independently) in 1896. It is legend that everyone who finds a new proof
will become one hundred years old and, indeed, both Hadamard and de la
Vallée-Poussin lived almost a century. The aim of this section is to prove
Theorem 1.9. There exists a positive constant c such that, for x ≥ 2,
1
.
π(x) = Li(x) + O x exp −c(log x) 9
The integral logarithm can be approximated by x/ log x; however, this is a
less good approximation to π(x) as the following table illustrates.
x
π(x)
Li(x)
3
10
168
178
6
78498
78628
10
9
10
50847534
50849235
12
37607912018 37607950281
10
error in %
x/ log x
5.95
145
0.1656
72382
0.003345
48254942
0.0001017 36191206825
error in %
14
7.8
5.1
3.8
Out of technical reasons we prefer to work with the logarithmic derivative
of ζ(s) (instead of log ζ(s) as Riemann did). Logarithmic differentiation of
the Euler product (1.8) gives for σ > 1
∞
X
Λ(n)
ζ′
(s) = −
,
ζ
ns
n=1
where
log p if n = pk ,
0
otherwise,
is the von Mangoldt Λ-function. Since ζ(s) does not vanish in the half-plane
σ > 1, the logarithmic derivative is analytic for σ > 1. As we shall see below
all desired information on π(x) is encoded in
1
X
X
(1.27)
ψ(x) :=
Λ(n) =
log p + O x 2 .
Λ(n) :=
n≤x
p≤x
The idea of proof is simple. Partial summation gives
Z ∞
ζ′
ψ(x)
(1.28)
− (s) = s
dx.
ζ
xs+1
1
If we could transform this into a formula in which ψ(x) is isolated and given
in terms of a complex integral over the zeta-function, then we might hope to
find an asymptotic formula for ψ(x) by contour integration methods. Indeed,
such a transformation exists (Perron’s formula); however, this alone is not
sufficient. In order to prove Gauss’ conjecture we shall also need knowledge
Section 1.4
The prime number theorem
25
on the analytic behaviour of the zeta-function on and in neighbourhood of
the line σ = 1.
1.4.1. A zero-free region. First of all we shall establish a zero-free
region for ζ(s) which covers the abscissa of absolute convergence σ = 1. In
this delicate problem we follow (with slight modifications) the ideas of de La
Vallée-Poussin (see also Titchmarsh [200]).
In the sequel we shall only argue for s = σ + it from the upper half-plane;
with regard to ζ(s) = ζ(s) all estimates below can be reflected with respect
to the real axis.
Lemma 1.10. For t ≥ 8, 1 − 21 (log t)−1 ≤ σ ≤ 2,
ζ(s) ≪ log t
and
ζ ′ (s) ≪ (log t)2 .
Proof. Let 1 − (log t)−1 ≤ σ ≤ 3. If n ≤ t, then
1
s
σ
1−(log t)−1
log n ≫ n.
|n | = n ≥ n
= exp
1−
log t
Thus, (1.11) implies
ζ(s) ≪
X1
+ t−1 ≪ log t.
n
n≤t
The estimate for ζ ′(s) follows immediately from Cauchy’s formula
I
ζ(z)
1
′
dz,
ζ (s) =
2πi
(z − s)2
where the integration is taken over the circle |z−s| = 12 (log t)−1 ; alternatively,
one can perform (carefully) differentiation of (1.11). •
In view of the Euler product (1.8) we have, for σ > 1,
|ζ(σ + it)| = exp(Re log ζ(s)) = exp
X cos(kt log p)
p,k
kpkσ
!
.
Since
(1.29)
17 + 24 cos α + 8 cos(2α) = (3 + 4 cos α)2 ≥ 0,
it follows that
(1.30)
ζ(σ)17 |ζ(σ + it)|24 |ζ(σ + 2it)|8 ≥ 1.
This inequality is the main idea for our following observations. In view of
the simple pole of ζ(s) at s = 1 we have for small σ > 1
ζ(σ) ≪
1
.
σ−1
Chapter 1
26
Classical L-functions
Assuming that ζ(1 + it) has a zero for t = t0 6= 0, it would follow that
|ζ(σ + it0 )| ≪ σ − 1,
leading to
lim ζ(σ)17|ζ(σ + it0 )|24 = 0,
σ→1+
in contradiction to (1.30). Thus, the zeta-function has no zeros on the 1-line:
ζ(1 + it) 6= 0
for t ∈ R.
Actually, this non-vanishing argument should be compared with Mertens’
proof of L(1, χ) 6= 0. It can be shown that the non-vanishing of ζ(1 + it) is
equivalent to Gauss’ conjecture (1.2), i.e., the prime number theorem without error term, and we shall prove this equivalence in the following section.
However, here we are interested in a prime number theorem with error term.
For this purpose we have to enter the critical strip.
A simple refinement of the argument allows a lower estimate for the absolute
value of ζ(1+it): for t ≥ 1 and 1 < σ < 2, we deduce from (1.30) and Lemma
1.10
17
1
1
17
1
≤ ζ(σ) 24 |ζ(σ + 2it)| 3 ≪ (σ − 1)− 24 (log t) 3 .
|ζ(σ + it)|
Furthermore, with Lemma 1.10,
Z
(1.31) ζ(1 + it) − ζ(σ + it) = −
σ
1
ζ ′ (u + it) du ≪ |σ − 1|(log t)2 .
Hence
|ζ(1 + it)| ≥ |ζ(σ + it)| − c1 (σ − 1)(log t)2
17
1
≥ c2 (σ − 1) 24 (log t)− 3 − c1 (σ − 1)(log t)2 ,
where c1 , c2 are certain positive constants. Choosing a constant B > 0 such
17
that A := c2 B 24 − c1 B > 0 and putting σ = 1 + B(log t)−8 , we obtain
|ζ(1 + it)| ≥
(1.32)
A
.
(log t)6
This lower bound we shall use for an estimate to the left of the line σ = 1.
Lemma 1.11. We have
ζ(s) 6= 0
for
σ ≥ 1 − δ min{1, (log t)−8 };
more precisely, there exists a positive constant c3 such that
c3
.
(1.33)
|ζ(σ + it)| ≥
(log t)6
Section 1.4
The prime number theorem
27
Proof. In view of Lemma 1.10 estimate (1.31) holds for 1 − δ(log t)−8 ≤ σ ≤
1. Using (1.32), it follows that
|ζ(σ + it)| ≥
A − c1 δ
,
(log t)6
where the right-hand side is positive for sufficiently small δ. This yields
Lemma 1.11. •
The largest known zero-free region for the zeta-function was found by Vinogradov [204] and Korobov [121] (independently). Using Vinogradov’s ingenious method for exponential sums, they proved
c
(1.34)
ζ(s) 6= 0 in σ ≥ 1 −
2
1
(log |t|) 3 (log log |t|) 3
for some positive constant c and sufficiently large |t|; for a proof see Ivić [98].
However, it is still unknown whether there exists any ǫ > 0 such that ζ(s)
does not vanish for σ > 1 − ǫ. No progress here for almost half a century!
1.4.2. Perron’s formula. The next ingredient in the proof of the prime
number theorem is
Lemma 1.12. For positive real numbers c, y, T , define
Z c+iT s
y
1
ds
I(y, T ) =
2πi c−iT s
and
δ(y) =
Then
|I(y, T ) − δ(y)| <

 0
1
2

1
if
if
if
0 < y < 1,
y = 1,
y > 1.
y c min{1, (T | log y|)−1} if y 6= 1,
c/T
otherwise.
The expression δ(y) is a good approximation to the integral I(y, T ) since
Z c+i∞ s
1
y
(1.35)
I(y, ∞) = lim I(y, T ) =
ds = δ(y),
T →∞
2πi c−i∞ s
and the error term is rather small.
Proof. For y = 1 and s = c + it, we find
Z T
Z
Z
1
1 T
1 1 ∞ du
dt
c
I(1, T ) =
=
dt = −
,
2π −T c + it
π 0 c2 + t2
2 π T /c 1 + u2
where we have used the fact that
Z U
0
du
= arctan U
1 + u2
Chapter 1
28
Classical L-functions
and arctan U tends to π2 as U → ∞. Now it is easy to deduce the desired
estimate for |I(1, T ) − δ(1)|.
Now assume that 0 < y < 1 and r > c. Since the integrand is analytic in
σ > 0, Cauchy’s theorem implies, for T > 0,
Z r−iT Z r+iT Z c+iT s
1
y
I(y, T ) =
+
ds.
+
2πi
s
c−iT
r−iT
r+iT
For σ = r we have
|
Hence, as r → ∞,
I(y, T ) =
resp.
yr
1
ys
|≤
≤ .
s
r
r
1
−
2πi
Z
∞+iT
c+iT
1
+
2πi
Z
∞−iT c−iT
ys
ds,
s
Z ∞
yc
1
y σ dσ ≤
.
|I(y, T )| ≤
πT c
T | log y|
This is the estimate for 0 < y < 1.
Finally, if y > 1, then we integrate over the rectangular contour with
corners c ± iT, −r ± iT , analogously. In this case the pole of the integrand
at s = 0 with residue
ys
ys
Res s=0 = lim · s = 1
s→0 s
s
gives the value δ(y) = 1 for the integral in question; the error estimate follows
as in the previous case. •
We apply this lemma to the logarithmic derivative of the zeta-function.
Lets assume that x 6∈ Z and c > 1. Then
Z c+i∞ X
Z c+i∞ s
∞
∞
X
Λ(n) xs
ds
x
ds =
Λ(n)
;
s
s
n
s
c−i∞ n=1 n
c−i∞
n=1
here interchanging integration and summation is allowed by the absolute
convergence of the series. In view of Lemma 1.12 with T → ∞ (i.e., (1.35))
it follows that
Z c+i∞ X
∞
X
1
Λ(n) xs
Λ(n) =
ds,
2πi c−i∞ n=1 ns s
n≤x
resp.
(1.36)
1
ψ(x) =
2πi
Z
c+i∞ c−i∞
s
ζ′
x
− (s)
ds.
ζ
s
This is Perron’s formula and, of course, it holds in a far more general setting
for arbitrary Dirichlet series in the half-plane of absolute convergence. However, for applications it is often useful to work with integrals over compact
Section 1.4
The prime number theorem
29
line segments. Lemma 1.12 yields
1
ψ(x) = −
2πi
Z
c+iT
c−iT
ζ ′ xs
(s) ds + error(x, T, c),
ζ
s
where
∞
xc X Λ(n)
error(x, T, c) ≪
.
T n=1 nc | log nx |
We split the series on the right-hand side as follows

 X

|n−x|> x
4
+

X 
|n−x|≤ x
4

Λ(n)
.
nc | log nx |
Since | log nx |−1 is bounded by a constant in the first sum and ≪ log x in the
second one, we get
(1.37)
Z c+iT ′
xs
ζ
1
(s) ds +
ψ(x) = −
2πi c−iT ζ
s
c ′ x(log x)2
x ζ
(c) +
+ log x .
+O
T ζ T
1.4.3. Final steps of the proof. Now we are in the position to prove
Gauss’ conjecture (1.2), the celebrated prime number theorem. Here we shall
combine our observation from the previous two sections.
In order to find an asymptotic formula for the integral in (1.37) we move
the path of integration to the left. By the theorem of residues we shall obtain
contributions from the poles of the integrand, i.e.,
• the zeros of ζ(s) inside the contour,
• the pole of ζ(s) at s = 1, and
s
• the pole of xs at s = 0 (if surrounded by the contour);
the latter quantity is independent of x and therefore a constant. For our
purpose it is sufficient to include only the pole at s = 1; however, later, when
we are going to prove the explicit formula, we have to include all appearing
poles. In view of the zero-free region of Lemma 1.11 we put c = 1 + λ with
λ = δ(log T )−8 , where δ is given by Lemma 1.11, and integrate over the
boundary of the rectangle R given by the corners 1 ± λ ± iT . By this choice
ζ(s) does not vanish in and on the boundary of R. The calculus of residues
Chapter 1
30
Classical L-functions
implies
c+iT
′ s
x
ζ
ds
− (s)
ζ
s
c−iT
Z 1−λ−iT Z 1−λ+iT Z 1+λ−iT ′ s
ζ
x
=
+
+
− (s)
ds
ζ
s
1+λ−iT
1−λ−iT
1−λ+iT
′ s
ζ
x
+2πiRes s=1 − (s)
.
ζ
s
Z
For the logarithmic derivative of ζ(s) we have
d
1
ζ′
log ζ(s) =
+ O(1)
− (s) = −
ζ
ds
s−1
as s → 1. Thus, we obtain for the residue at s = 1
′ s
s
ζ
x
1
x
Res s=1 − (s)
= lim(s − 1) ·
+ O(1)
= x;
s→1
ζ
s
s−1
s
this will turn out to be the main term. It remains to bound the integrals.
For the horizontal integrals we find with regard to Lemma 1.11
Z 1+λ±iT ′ s
x
x1+λ
ζ
ds ≪
.
− (s)
ζ
s
T
1−λ±iT
Further, for the vertical integral,
Z 1−λ+iT ′ s
x
ζ
ds ≪ x1−λ (log T )9 .
− (s)
ζ
s
1−λ−iT
Collecting together, we deduce from (1.37)
1+λ
x
x(log x)2
1−λ
9
ψ(x) = x + O
+ x (log T ) +
+ log x .
Tλ
T
1
1
Choosing T = exp(δ 10 (log x) 9 ), we arrive at
1
ψ(x) = x + O x exp(−c(log x) 9 )
for some positive constant c. Now it easily follows from (1.27) that also
X
1
(1.38)
θ(x) :=
log p = x + O x exp(−c(log x) 9 ) .
p≤x
Applying partial summation, we find
X
1
π(x) =
log p ·
log p
p≤x
Z x
θ(x)
d 1
=
−
du
θ(u)
log x
du log u
2
Z x
1
d 1
x
9
.
−
du + O x exp −c(log x)
u
=
log x
du log u
2
Section 1.4
The prime number theorem
31
Now partial integration shows that the first two terms on the right-hand side
are equal to the integral logarithm (up to a constant); this finishes the proof
of the prime number theorem 1.9. •
Reviewing the proof we see that the simple pole of the zeta-function is not
only the key in Euler’s proof of the infinitude of primes but also gives the
main term of the asymptotic formula in the prime number theorem.
In view of the largest known zero-free region (1.34) one can obtain the
following stronger form of the prime number theorem:
!!
3
5
(log x)
.
(1.39)
π(x) = Li(x) + O x exp −c
1
(log log x) 5
1.4.4. A probabilistic model and its limits. The prime numbers,
which – on first sight – seem to be randomly distributed among the positive
integers, satisfy a strong distribution law! The prime number theorem allows
the following probabilistic interpretation: the probability that a given positive
integer n is prime is (asymptotically) equal to log1 n . We may use this interpretation in order to make some heuristics about prime numbers of a special
shape.
The Mersenne numbers are given by Mp = 2p − 1, where p is prime; notice that if the exponent p is not prime, one can easily factor 2p − 1. For
the Mersenne numbers there exist a very simple (and fast) primality test.
Consider the following iteration
s := 4, for i from 3 to p do s := s2 − 2 mod (2p − 1).
The Lucas-Lehmer test states that Mp is prime if and only if the iteration
yields the result s = 0 (the test is simple; however, its proof is rather involved;
see [81]). The sequence of iterated values of s (not reduced mod Mp ) starts
with
s=4
7→
14 = 2 · 7
7→
194
7→
37 634 = 2 · 31 · 607,
from which we can read the first two Mersenne primes 7 and 31.1 It is
unknown whether there are infinitely many Mersenne primes; however, we
might be optimistic: using the probabilistic model, a number Mp is prime
with probability
1
1
∼
,
log Mp
p log 2
1The
currently largest known prime number is a Mersenne prime, naemly M30 402 457
found by Cooper & Boone in December 2005 (see http://www.mersenne.org/prime.htm for
its 9 152 052 digits and the Great Internet Mersenne Prime Search, initiated by Woltman).
32
Chapter 1
Classical L-functions
and hence the expectation value for the number of Mersenne primes is
1 X1
,
log 2 p p
which is divergent.
In the 1920s, Hardy & Littlewood developed some heuristics for more advanced questions. We illustrate their reasoning with a famous open problem.
Two numbers p and p + 2 are said to be twin primes if both p and p + 2
are prime numbers. It is a long-standing conjecture that there are infinitely
many twin primes. Hardy & Littlewood [80] gave a conjectural asymptotic
formula for the number of twin primes as follows. According to our probabilistic model we observe: given that n is prime, if one is supposed that n + 2
to be random, its chance of being prime would be
1
1
∼
log(n + 2)
log n
too, and so the probability of primality of both n and n+2 would be (log n)−2 .
However, if n is prime, then n + 2 can fall into n − 1 residue classes mod p
for any prime p 6= n, of which p − 2 are non-zero. Thus the chance that p
does not divide n + 2 is (p − 2)/(p − 1) rather than (p − 1)/p as it would be
if n + 2 were random. Hence, we have to expect a correction factor
1
(p − 2)/(p − 1)
=1−
(p − 1)/p
(p − 1)2
for each odd prime p; clearly, the oddest prime p = 2 is not a twin. Since
half of the integers is odd but with n also n + 2 is odd, we further have to
multiply with a factor 2. Hence, it is natural to conjecture that the number
of twin primes n, n + 2 with n ≤ x is asymptotically equal to
Y
x
1
1−
2
2
(p − 1) (log x)2
p6=2
as x → ∞. Computations support this conjecture. By his extension of Eratosthenes’s sieve method, Brun [27] showed that the number of twin primes
below x is bounded above by O(x/(log x)2 ) which implies the convergence of
the series over the reciprocals of twin primes:
X 1
< ∞,
p
p≤x
p+2 prime
in contrast to the divergence of the sum of reciprocals of all primes. Brun’s
result indicates that almost all primes are not twin primes. A more general
conjecture is the one of Bateman & Horn [13] on prime values of polynomials
which seems to be far out of reach with present day methods.
Section 1.4
The prime number theorem
33
In 1936, Cramér [42] introduced the following model for the distribution
of prime numbers:
“Let U1 , U2 , U3 , . . . be an infinite series of urns containing black
and white balls, the chance of drawing a white ball from Un
being log1 n for n > 2 while the composition of U1 and U2 may
be arbitrarily chosen. We now assume that one ball is drawn
from each urn, so that an infinite series of alternately black
and white balls is obtained. If Pn denotes the number of the
urn from which the nth white ball in the series was drawn, the
numbers P1 , P2 , . . . will form an increasing sequence of integers,
and we shall consider the class C of all possible sequences (Pn ).
Obviously the sequence S of prime numbers (pn ) belongs to this
class. We shall denote by Π(x) the number of those Pn which
are ≤ x, thus forming an analogy to the ordinary notation π(x)
for the number of primes pn ≤ x. (. . .) As a matter of fact, it
may be shown that, with probability 1, the relation
|Π(x) − Li (x)|
q
=1
lim sup √
x→∞
2x · logloglogx x
is satisfied. With respect to the corresponding difference π(x) −
Li (x) in the prime number problem, it is known that, if the
Riemann hypothesis is assumed, the true√maximum order of
√
this difference lies between the functions logxx and x log x. It
is interesting to find that the order of the function occurring in
the denominator in the above equation falls inside this interval
of indetermination.”
Cramér used this model in order to conjecture an asymptotic formula for
the largest gap between consecutive primes. Denote by pn the nth prime in
ascending order. Cramér was led to conjecture that
max(pn+1 − pn ) ∼ (log x)2 .
pn ≤x
This seems to be a good guess but only little is known in this direction.
Recently, a related problem was solved by Goldston, Pintz & Yildirim [65].
They showed that there exist
pn+1 − pn
= 0.
lim inf
n→∞
log pn
Their method depends on the level of distribution of primes in arithmetic
progressions. Assuming the Elliott-Halberstam conjecture (which is to complicated to be given here), they also proved that there are infinitely often
primes differing by 16 or less. This is a remarkable progress towards the twin
prime conjecture!
34
Chapter 1
Classical L-functions
Another question in prime number distribution theory asks for which functions Φ(x) does
Φ(x)
π(x + Φ(x)) − π(x) ∼
log x
7
hold as x → ∞? Huxley [92] proved that one can choose Φ(x) = x 12 +ǫ ;
under assumption of the Riemann hypothesis one can replace the exponent
by 12 + ǫ. Assuming Riemann’s hypothesis, Selberg [178] proved that the
asymptotic formula is true for almost all values of x provided Φ(x)/(log x)2
tends with x to infinity; here the notion for almost all values of x means
that the set of exceptional x ≤ u has Lebesgue measure o(u) as u → ∞.
Cramér’s probabilistic model predicts that one can relax Selberg’s condition
to Φ(x) ≫ (log x)2 . However, this was disproved by Maier [140] who showed
by his celebrated matrix method that
π(x + Φ(x)) − π(x)
π(x + Φ(x)) − π(x)
>1
and
lim inf
< 1,
lim sup
x→∞
Φ(x)/ log x
Φ(x)/ log x
x→∞
where Φ(x) = (log x)λ for λ > 1. Hence, the local distribution of primes does
not follow this simple probabilistic model. This observation started the search
for further violations of Cramér’s model as well as for suitable modifications.
A nice survey on this topic is Granville [66].
Exercise 28. Prove (1.27) and (1.38). Further, fill the gaps left in the proof of
Lemma 1.12.
Exercise 29. Try other trigonometric identities like (1.29) in order to obtain a
better error term in the prime number theorem.
Exercise 30. Prove that if pn is the n-th prime number, then
pn ∼ n log n.
Show that this also implies the prime number theorem without error term.
Exercise 31. Prove a Perron type formula for arbitrary Dirichlet series in their
half-plane of absolute convergence.
Exercise 32. * Prove the following prime number theorem for arithmetic progressions: Let a and q be fixed positive and coprime integers. Then there exists a
positive constant c, depending only on a mod q, for which
1
1
π(x, a mod q) =
Li (x) + O x exp −c(log x) 9
.
ϕ(q)
Exercise 33. * Denote by dk (n) the number of representations of the positive
integer n as a product of k positive integers.
i) Show that, for σ > 1,
∞
X
dk (n)
k
.
ζ(s) =
ns
n=1
Section 1.5
Tauberian theorems
35
ii) Prove that
X
dk (n) = xPk (log x) + error,
n≤x
s
where Pk (X) is a polynomial of degree k − 1 in X, equal to Res s=1 ζ(s)k xs and
the error term is reasonably small. In the case k = 2 compare with the result of
Exercise 21 i).
k
The numbers Fk = 22 + 1 with k = 0, 1, 2, . . . , are called Fermat numbers.
Gauss showed that the regular n-gon can be constructed only by use of ruler and
compass if and only if n is a power of 2 times a product of distinct prime Fermat
numbers.
Exercise 34. Compute the first 10 Fermat numbers and test whether they are
prime. Using the heuristics of the previous section, state a conjecture on the number of prime Fermat numbers.
The famous open Goldbach conjecture claims that any even integer greater than
or equal to 4 can be written as a sum of two prime numbers.
Exercise 35. Use the Hardy & Littlewood heuristics
i) to find an asymptotic formula for the number of representations of a large positive
integer as a sum of two primes, and
ii) to state a conjectural asymptotic formula for the number of primes of the form
p = n2 + 1 below x.
Recently, Green & Tao [69] proved a famous conjecture, namely, that the set of
prime numbers contains arbitrarily long arithmetic progressions (see also Green’s
survey [68]).
Exercise 36. * Use the probabilistic model in the form
Prob.(n is prime : 1 ≤ n ≤ N )
to show that
N2
,
(log N )k
under the assumption that the events that n + jd and n + ℓd for j 6= ℓ are all
independent. Can you make the asymptotics more precise by a reasoning a la
Hardy & Littlewood?
Expect. (♯{1 ≤ n, d ≤ N : n, n + d, . . . , n + (k − 1)d are all prime}) ≍
1.5. Tauberian theorems – a general approach
In number theory we are often faced with the following problem: given a
sequence of complex numbers a(n), we want to know the behaviour of the
P
summatory function n≤x a(n). The prime number theorem may be a first
example and thus one may try to study the generating Dirichlet series by use
of Perron’s formula. This approach can be streamlined and the output can
Chapter 1
36
Classical L-functions
be found in so-called Tauberian theorems, developed by Hardy & Littlewood,
Ikehara, Wiener, and many others. A good overview on this rich theory gives
Korevaar [120].
1.5.1. The theorem of Wiener-Ikehara. Abel proved that
P∞
P∞
n
n=0 a(n) = 1 implies that
n=0 a(n)x tends to 1 as x → 1−. In 1897,
Tauber [196] proved that the converse implication holds if na(n) = o(1).
After Tauber plenty of similar results were proven, many of them with direct
applications to number theory (created with number theoretical motivation
in mind). Following Bochner, resp. Chandrasekharan [29] here we shall
prove the Tauberian theorem of Ikehara [95] and Wiener [212]:
Theorem 1.13. Let A(x) be a non-negative, non-decreasing function of x ∈
[0, ∞). Suppose that the integral
Z ∞
A(x) exp(−sx) dx
0
converges to the function f (s) and that f (s) is analytic in the half-plane
σ ≥ 1, except for a simple pole at s = 1 with residue 1. Then
lim A(x) exp(−x) = 1.
x→∞
Proof. Define B(x) = A(x) exp(−x). First we shall prove that, for any
positive λ,
2
Z λy v sin v
dv = π.
(1.40)
lim
B y−
y→∞ −∞
λ
v
For σ > 1, we have
Z ∞
f (s) =
A(x) exp(−sx) dx
and
0
Thus,
1
=
F (s) := f (s) −
s−1
Z
0
∞
1
=
s−1
Z
0
∞
exp((1 − s)x) dx.
(B(x) − 1) exp((1 − s)x) dx.
By assumption F (s) is analytic for σ ≥ 1. Now define Fǫ (t) = F (1 + ǫ + it)
for ǫ > 0. For λ > 0, we obtain
Z 2λ
|t|
exp(iyt) dt
Fǫ (t) 1 −
2λ
−2λ
Z 2λ |t|
(1.41)
exp(iyt) ×
=
1−
2λ
−2λ
Z ∞
×
(B(x) − 1) exp(−(ǫ + it)x) dx dt.
0
Section 1.5
Tauberian theorems
37
Next we want to interchange the order of integration on the right-hand side.
Since A(x) is non-negative and non-decreasing, for real s and x > 0,
Z ∞
A(x) exp(−sx)
f (s) ≥ A(x)
exp(−su) du =
,
s
x
resp. A(x) ≤ sf (s) exp(sx). Since f (s) is analytic for σ > 1, this implies
A(x) = O(exp(sx)) for any s > 1 and
B(x) exp(−δx) = A(x) exp(−(1 + δ)x) = o(1)
for every δ > 0. It follows that the integral
Z ∞
(B(x) − 1) exp(−(ǫ + it)x) dx
0
converges uniformly for −2λ ≤ t ≤ 2λ. Thus, we can interchange the order
of integration in (1.41) and obtain
Z 2λ
Z ∞
|t|
dt dx.
(B(x) − 1) exp(−ǫx)
exp(i(y − x)t) 1 −
2λ
0
−2λ
This leads with (1.41) to
Z 2λ
|t|
exp(iyt) dt
Fǫ (t) 1 −
2λ
−2λ
Z ∞
(sin(λ(y − x)))2
= 2
(B(x) − 1) exp(−ǫx)
(1.42)
dx.
λ(y − x)2
0
Since F (s) is analytic in σ ≥ 1, it follows that Fǫ (t) tends to F (1 + it) as
ǫ → 0, uniformly for −2λ ≤ t ≤ 2λ. Moreover,
Z ∞
Z ∞
(sin(λ(y − x)))2
(sin(λ(y − x)))2
dx
=
dx.
lim
exp(−ǫx)
ǫ→0 0
λ(y − x)2
λ(y − x)2
0
We deduce
Z ∞
Z ∞
(sin(λ(y − x)))2
(sin(λ(y − x)))2
lim
B(x) exp(−ǫx)
dx
=
dx.
B(x)
ǫ→0 0
λ(y − x)2
λ(y − x)2
0
By (1.42),
(1.43)
1
2
2λ
|t|
exp(iyt) dt
F (1 + it) 1 −
2λ
−2λ
Z ∞
(sin(λ(y − x)))2
dx.
=
(B(x) − 1)
λ(y − x)2
0
Z
The Riemann-Lebesgue lemma states that
Z ∞
lim
f (x) exp(ixy) dx = 0
y→∞
−∞
Chapter 1
38
Classical L-functions
for any absolutely integrable function f . Thus, letting y → ∞, the left-hand
side of (1.43) tends to zero while
2
Z λy Z ∞
sin v
(sin(λ(y − x)))2
dv = π.
dx = lim
(1.44)
lim
y→∞ −∞
y→∞ 0
λ(y − x)2
v
Hence,
lim
y→∞
Z
∞
B(x)
0
(sin(λ(y − x)))2
dx = π;
λ(y − x)2
this proves (1.40).
In order to prove the theorem we have to show
(1.45)
1 ≤ lim inf B(x) ≤ lim sup B(x) ≤ 1.
x→∞
x→∞
Clearly, this implies the existence of the limit limx→∞ B(x) and that this
limit is equal to 1. For given positive numbers a and λ let y > λa . By (1.40),
2
Z a v sin v
dv ≤ π
lim sup
B y−
λ
v
y→∞
−a
(the integrand is non-negative). Since A(u) = B(u) exp(u) is non-decreasing,
we have, for −a ≤ v ≤ a,
a
a
v
v
B y−
exp y −
≤B y−
exp y −
.
λ
λ
λ
λ
This implies
v
a
v−a
a
2a
B y−
≥B y−
exp
≥B y−
exp −
.
λ
λ
λ
λ
λ
Hence,
Z a 2
2a
sin v
a
exp −
dv
lim sup B y −
λ
λ
v
y→∞
−a
2
Z a v sin v
dv ≤ π.
= lim sup
B y−
λ
v
y→∞
−a
For fixed a and λ we have lim supy→∞ B(y − λa ) = lim supy→∞ B(y). Thus,
2
Z a
2a
sin v
exp −
lim sup B(y)
dv ≤ π,
λ
v
y→∞
−a
being valid for all a > 0 and λ > 0. Letting a, λ → ∞ such that
deduce
2
Z ∞
sin v
dv ≤ π.
lim sup B(y)
v
y→∞
−∞
a
λ
→ 0, we
Section 1.5
Tauberian theorems
39
Now (1.44) implies the desired upper bound for lim supy→∞ B(y). The just
proved inequality yields the existence of a constant c such that |B(x)| ≤ c.
Hence, for fixed positive a and λ and a sufficiently large y,
2
Z λy v sin v
dv
B y−
λ
v
−∞
2
2
Z −a Z ∞ Z a v sin v
sin v
(1.46) ≤ c
dv +
dv.
B y−
+
v
λ
v
−a
−∞
a
As above, we have B(y − λv ) ≤ B(y + λa ) exp( 2a
) for −a ≤ v ≤ a. Therefore,
λ
2
Z a 2
Z a v sin v
2a
sin v
a
B y−
exp
dv ≤ B y +
dv.
λ
v
λ
λ
v
−a
−a
From (1.40), (1.46) and the latter inequality it follows that
Z −a Z ∞ 2
sin v
π ≤ c
+
dv +
v
−∞
a
Z a 2
2a
sin v
a
exp
dv.
+ lim inf B y +
y→∞
λ
λ
v
−a
Here we may replace lim inf y→∞ B(y + λa ) by lim inf y→∞ B(y). Then, after
sending a, λ → ∞ such that λa → 0, we get the desired lower bound for
lim inf y→∞ B(y). The theorem is proved. •
Now we shall derive a reformulation of Theorem 1.13 to which we also refer
to as the Theorem of Wiener & Ikehara:
P
−s
be a Dirichlet series with nonTheorem 1.14. Let F (s) = ∞
n=1 a(n)n
negative real coefficients and absolutely convergent for σ > 1. Assume that
F (s) can be extended to a meromorphic function in σ ≥ 1 such that there are
no poles except for a possible simple pole at s = 1 with residue r ≥ 0. Then
X
A(x) :=
a(n) = rx + o(x).
n≤x
Proof. Without loss of generality we may suppose that the residue is positive: r > 0, since otherwise we can consider the function F (s) + ζ(s) (which
then has residue r + 1 = 1). Furthermore, we may assume that r = 1 simply
by replacing a(n) by a(n)/r.
By partial summation (as we did for zeta in the proof of Theorem 1.3),
Z ∞
A(x)
dx,
F (s) = s
xs+1
1
resp.
Z ∞
F (s)
=
A(exp(y)) exp(−ys) dy
s
0
Chapter 1
40
Classical L-functions
with x = exp(y). Now in view on all assumptions on F (s) it follows from
Theorem 1.13 that
lim A(exp(y)) exp(−y) = 1.
y→∞
Re-substituting x = exp(y) we get the assertion. •
1.5.2. The prime number theorem for arithmetic progressions
- revisited. As first application of the just proven Tauberian theorem we
return to the question how the prime numbers are distributed in the residue
classes. It is natural to ask for a quantitative version of Dirichlet’s prime
number theorem for arithmetic progressions. Here we shall prove (1.23).
Let χ mod q be a character. Similarly as for the Riemann zeta-function, we
consider the logarithmic derivative of a Dirichlet L-functions L(s, χ), given
by
∞
X
Λ(n)χ(n)
L′
(s, χ) = −
,
s
L
n
n=1
where Λ(n) is the von Mangoldt-function (introduced in the previous section).
We define
X
X
Λ(n).
ψ(x; χ) =
Λ(n)χ(n)
and
ψ(x; a mod q) =
n≤x
n≤x
n≡a mod q
By the orthogonality relation for characters (1.17), we find
X
1
ψ(x; a mod q) =
χ(a)ψ(x; χ).
ϕ(q) χ mod q
Now suppose that a and q are coprime (otherwise the functions in the latter
identity are all bounded). We want to apply Theorem 1.14 with the functions
X
L′
F (s) = −
and
A(x) = ψ(x; a mod q).
χ(a) (s, χ)
L
χ mod q
Notice that the left-hand side has a Dirichlet series representation for σ > 1
with non-negative coefficients. From Section 1.3 we know that L(s, χ) is
analytic for σ ≥ 1 if χ is not the principal character. In the case of the
principal character we find, by (1.19),
−
1
L′
(s, χ0 ) =
+ higher terms.
L
s−1
Finally, we have to assure that any of the appearing L(s, χ) has no zero on the
1-line. By Theorem 1.7 we already know that L(1, χ) 6= 0. For an arbitrary
point s = 1 + it with t 6= 0 one may argue as we did for the zeta-function in
the previous section just by replacing (1.30) by
L(σ, χ0 )17 |L(σ + it)|24 |L(σ + 2it, χ2 )|8 .
Section 1.5
Tauberian theorems
41
Thus, applying Theorem 1.14 we obtain ψ(x; a mod q) ∼ ϕ(q)−1 x. By partial
summation, this implies
Theorem 1.15. Let a and q be coprime integers. Then, as x → ∞,
1
x
π(x; a mod q) ∼
.
ϕ(q) log x
We did not use any information about the behaviour of the involved Dirichlet
L-functions from inside the critical strip. Therefore, we do not get an error
term. One can easily prove an asymptotic formula with error term following
the argument we gave for zeta; however, for applications one often wants to
have a result which is uniform in the modulus; for instance, for bounds of
the least prime in an arithemtic progression. The theorem of Page-SiegelWalfisz provides such an asymptotic formula which is uniform in a small
region of values q. In this case the situation is more delicate than for the
zeta-function of fixed modulus. In principle, one cannot exclude that certain
L(s, χ) have real zeros on the real axis inside the critical strip. These socalled exceptional zeros (or Siegel zeros) are difficult to deal with. We shall
not go into the details here but refer to Prachar [172].
1.5.3. Dedekind zeta-functions and the prime ideal theorem. Before we can introduce this further class of L-functions we have to recall some
basic facts from algebraic number theory. A good reading on this topic is
Swinnerton-Dyer [195].
A complex number α is said to be algebraic over Q if there exists a nonzero polynomial P (X) with integer coefficients such that P (α) = 0; the
polynomial with coprime coefficients and least degree having this property is
called the minimal polynomial of α and is denoted by Pα . The degree of the
minimal polynomial is said to be the degree of α. For algebraic α, the set
Q[X]/Pα (X)
is a finite algebraic extension of the field of rational numbers, the algebraic
number field associated to α; a more convenient form is
Q(α) := a0 + a1 α + . . . + ad−1 αd−1 : aj ∈ Q ,
where d = deg α = deg Pα . The degree of the field extension Q(α)/Q is
equal to the dimension of the field Q(α) as a Q-vector space, and we write
d = [Q(α) : Q]. Note that any number in Q(α) is algebraic of degree ≤ d.
The zeros α1′ , . . . , αd′ of the minimal polynomial Pα (X) are the conjugates of
α (in fact, they are the images of α under the field automorphisms) and have
degree equal to d = deg α. Denote by σ1 , . . . , σ d the embeddings of K in C.
Then the discriminant of K is given by
dK = det ((σi (αj ))1≤i,j≤d )2 .
42
Chapter 1
Classical L-functions
In the case of a quadratic number field, i.e., [Q(α) : Q] = 2, this discriminant
is equal to the discriminant of the minimal polynomial Pα (X). The product
of all conjugates is the norm of α:
N(α) :=
d
Y
αj′ .
j=1
The norm provides a measure for the size of algebraic numbers. An algebraic
number is said to be an algebraic integer if its minimal polynomial is monic; in
this case, the norm is, up to the sign, equal to the constant term Pα (0) ∈ Z
in the minimal polynomial. The notion of algebraic integers extends the
standard notion of integers from Q to number fields. In fact, one can show
that an algebraic integer, which is rational, is a rational integer. The set of
all algebraic integers in a number field K forms a ring OK , the so-called ring
of integers. Unfortunately, these rings in general do not have a unique prime
factorization. For example, the identity
√
√
2 · 3 = (1 − −5) · (1 + −5)
√
gives two distinct factorizations of 6 in the ring Z[ −5] into irreducible factors. In order to obtain unique factorization we have to pass to ideals.
An ideal a of OK is a set of integers in OK having the properties
• if α, β ∈ a, then α + β ∈ a,
• if α ∈ a and λ ∈ OK , then λα ∈ a.
If a is a non-zero ideal of OK , then OK /a is a ring; its cardinality is denoted
by N(a) and is called the norm of a. An ideal a 6= (0) is said to be fractional
if there exists an integer α 6= 0 for which αa is an ideal of OK . An ideal a (not
necessarily fractional) is called principal if there exists α ∈ K if a = αOK .
A fractional ideal lies in OK if and only if it is an ideal of OK , in which case
we say that it is an integral ideal. In OK every fractional ideal is invertible,
i.e., a−1 a ⊂ OK . The set of fractional ideals forms a group. An ideal p of
OK is said to be a prime ideal if p 6= OK and if the quotient ring OK /p is an
integral domain, i.e., αβ ∈ p implies α ∈ p or β ∈ p. A non-zero prime ideal
is maximal. And, most importantly, every fractional ideal a has a unique
factorization into a product of powers of prime ideals.
We
√ shall give an example and consider quadratic number fields: K =
Q( D), where D is a squarefree integer.
√ It is not too difficult to show
that every rational prime p splits in Q( D) into prime ideals according to
the value of the Legendre symbol ( dpK ): a rational prime p is said to be
• inert: (p) = p if ( dpK ) = −1,
• ramified: (p) = p2 if ( dpK ) = 0,
• split: (p) = p1 p2 with p1 6= p2 if ( dpK ) = +1.
Section 1.5
Tauberian theorems
43
In the case of p = 2 we find 2 is inert if D ≡ 5 mod 8; otherwise
√
√

1
1
(1
−
D))((2,
D))
 (2, 2 (1 +
2
√ 2
(2) =
(2, 1√+ D)

(2, D)2
if D ≡ 1 mod 8,
if D ≡ 3, 7 mod 8,
if D ≡ 2, 6 mod 8.
Now we are in the position to introduce a new zeta-function which carries information about the arithmetic of number fields and is named after
Dedekind who set the foundations of ideal theory. The Dedekind zetafunction of a number field K over Q is given by
ζK (s) =
X
a
−1
Y
1
1
=
1−
,
s
N(a)s
N(p)
p
where the sum is taken over all non-zero integral ideals a and the product is
taken over all prime ideals p of the ring of integers of K. The identity between
series and product is an analytic version of the unique factorization of integral
ideals in prime ideals (analogously to the unique prime factorization of the
integers). Since the norm of an integral ideal is a positive rational integer,
the series can be rewritten as a Dirichlet series:
X
a
∞
X fK (n)
1
=
,
N(a)s
ns
n=1
where fK (n) counts the number of integral ideals a with N(a) = n. We
see that the Riemann zeta-function is the Dedekind zeta-function of Q and,
as a matter of fact, Dedekind zeta-functions share many properties with
Riemann’s zeta. First of all, we have to show that the Dirichlet series defining
the Dedekind zeta-function ζK (s) converges for σ > 1, independent of the field
K. To see this note that, for real s > 1,
−1 Y −d
Y
1
1
1−
|ζK (s)| = 1− σ
= ζ(σ)d ,
≤
p
N(p)s
p
p
since there are at most d = [K : Q] many primes p lying above each rational
prime p and N(p) is smallest if (p) splits
√ completely.
We return to our example K = Q( D). We write for short d := dQ(√D)
(since now there won’t be any confusion with the degree), which is equal to D
if D ≡ 1 mod 4, and equal to 4D if D ≡ 2, 3 mod 4. In view of the splitting
Chapter 1
44
Classical L-functions
of the primes one easily finds
ζQ(√D) (s)
Y =
( pd )=+1
×
(1.47)
1
1− s
p
Y ( pd )=−1
−2 Y −1
1
×
1− s
p
d
( p )=0
1
1− s
p
−1 1
1+ s
p
−1
= ζ(s)L(s, χ d),
with the Jacobi symbol, defined by
χ d : N → C,
Y
ν d
d
n 7→
=
,
n
p
j
j=1
where n = p1 · . . . · pν is the factorization of the integer n into prime factors
(not necessarily distinct).
In 1917, Hecke [86] obtained the first deeper results concerning the analytic
behaviour of Dedekind zeta-functions. He showed that (s−1)ζK (s) is an entire
function and that the Dedekind zeta-function has a simple pole at s = 1 with
residue
lim (s − 1)ζK (s) =
(1.48)
s→1+
2r1 (2π)r2 hR
p
,
ω |dK|
where r1 is the number of real conjugate fields, 2r2 is the number of complex
conjugate fields, h is the class number, R is the regulator, ω is the number
of roots of unity, and dK is the discriminant of K. We see there is a lot of
arithmetic information is encoded in this residue! The class number is the
number of equivalence classes of fractional ideals of K, and so it measures the
deviation of OK from having unique prime factorization. Gauss conjectured
that the√class numbers h = h(d) of an imaginary quadratic number field
K = Q( D) with discriminant d < 0 tend with −d to infinity; notice that
d = D if D ≡ 1 mod 4, and d = 4D if D ≡ 2, 3 mod 4. This was first
proved by Heilbronn [90] and in refined form by Siegel [183]. The problem
of finding an effective algorithm to determine all imaginary quadratic fields
with a given class number h is known as the Gauss class number h problem.
This problem is of interest with respect to the non-existence of exceptional
real zeros of Dirichlet L-functions off the critical line. The general Gauss
class number problem was solved by Goldfeld, Gross & Zagier [63, 71]. A
complete determination of the imaginary quadratic fields with class number 1
was first given by Heegner [89] (but his solution was not completely accepted
due to a number of gaps), Baker [10], and Stark [187] (independently):
h=1
⇐⇒
d ∈ {−3, −4, −7, −8, −11, −19, −43, −67, −163}.
Section 1.5
Tauberian theorems
45
Notice that class number 1 is equivalent to unique prime factorization in the
corresponding ring of algebraic integers. Further, note that (1.48) contains
the information of Dirichlet’s analytic class number formula.
We want to apply the theorem of Wiener & Ikehara 1.14. Again we consider
the logarithmic derivative:
−
X ΛK (a)
ζK′
(s) =
,
ζK
N(a)s
a
where
ΛK (a) :=
log N(p) if a = pk ,
0
otherwise.
Furthermore,
−
ζK′
1
(s) =
+ higher terms,
ζK
s−1
independent of the residue of ζK (s) at s = 1. Finally we have to assure that
ζK (s) has no zeros or further poles on the line σ = 1. One can show that the
Dedekind zeta-function of any number field K can be written as
Y
ζK (s) =
L(s, χ)δχ ,
χ
where the product is taken over so-called Artin L-functions (a class of Dirichlet L-functions we shall meet in Chapter 4) and the exponents δχ are integers
(not necessarily positive). However, in certain cases life is much easier: for instance, if K is a cyclotomic field, then this product is nothing but the product
of certain Dirichlet L-functions with positive exponents. Since L(1+it, χ) 6= 0
for all real numbers t and all characters χ, it then immediately follows that
ζK (s) does not vanish on the 1-line. However, this is true for any Dedekind
zeta-function. In general, one can prove the non-vanishing by another result
of Hecke (or Exercise 41). Now we can apply Theorem 1.14 and get
X
ψK (x) :=
ΛK (a) ∼ x.
N(a)≤x
Define
πK (x) = ♯{p ⊂ OK prime : N(p) ≤ x}.
By a standard application of partial summation we deduce the prime ideal
theorem:
Theorem 1.16. Let K be a number field. Then, as x → ∞,
x
.
πK (x) ∼
log x
46
Chapter 1
Classical L-functions
The first proof of the prime ideal theorem was given in 1903 by Landau
[124]. On the first view it might be surprising that the right-hand side does
not depend on the number field K. The residue of ζK (s) at s = 1 contains data
about the underlying field; however, the residue of the logartihmic derivative
of ζK (s) at s = 1 is equal to −1 independent of K.
We conclude with two remarks which are of special interest with respect to
our later studies. Hecke [86] proved that Dedekind zeta-functions satisfy a
functional equation:
!s
p
s r1
|dK |
Γ(s)r2 ζK (s)
Γ
n
2
2 r2 π 2
!1−s p
r
|dK |
1−s 1
=
Γ(1 − s)r2 ζK (1 − s).
(1.49)
Γ
n
r
2
2
2
2 π
It is further expected that also the analogue of the Riemann hypothesis is
true, i.e., all non-real zeros of ζK (s) lie on the critical line, resp. there are no
zeros for σ > 21 .
1.5.4. The prime number theorem and non-vanishing. Next we
shall prove that the prime number theorem without error term is equivalent
to the non-vanishing of ζ(s) on the line σ = 1. One implication follows
immediately from the theorem of Wiener-Ikehara (and can be proved just
along the lines of the previous applications of Theorem 1.14). To see the
other implication we assume that
(1.50)
ψ(x) ∼ x.
We have to deduce that there are no zeros of ζ(s) on the line σ = 1. For this
purpose define, for σ > 1,
Z ∞
ψ(x) − x
1
ζ ′(s)
−
=
dx.
Φ(s) = −
sζ(s) s − 1
xs+1
1
Clearly, Φ(s) is regular in σ > 0 except for simple poles at the zeros of the
zeta-function. It should be noticed that the logarithmic derivative has only
simple poles! Now (1.50) implies that, given ǫ > 0, there exists a real number
x0 such that for x > x0 we have |ψ(x) − x| < ǫx, and so we find for σ > 1
Z x0
Z ∞
|ψ(x) − x|
ǫ
ǫ
|Φ(s)| <
dx +
dx < C +
,
σ+1
σ
x
σ−1
1
x0 x
where C is a constant, depending only on ǫ. Hence,
(σ − 1)|Φ(σ + it)| < C(σ − 1) + ǫ.
Thus, for any fixed t, the limit of the left-hand side is 0 as σ → 1+. However,
if ζ(1+it) = 0 for some t 6= 0, then the limit of (σ−1)Φ(σ+it) would be equal
Section 1.5
Tauberian theorems
47
to the residue of Φ(s) at the simple pole s = 1 + it, and therefore different
from zero. Of course, the same reasoning applies to Dirichlet L-functions.
It was a big surprise when Erdös [50] and Selberg [179] obtained an elementary proof of the prime number theorem; here the attribute elementary
means that the proof does not use any arguments from analysis (apart from
simplest properties of the logarithm). Hence, the non-vanishing of ζ(s) for
σ ≥ 1 can be shown without complex analysis! The proofs of Erdös and
Selberg are not independent and their actual contributions are still under
discussion. (For the history of this quarrel read Goldfeld [64] and for an
elementary proof see [81].) In the meantime, even elementary proofs of the
prime number theorem with error term were given. Nevertheless, the analytic
approach yields more information on prime number distribution.
Exercise 37. Prove the prime number form in the form π(x) ∼
Theorem of Wiener & Ikehara 1.14.
x
log x
by using the
Exercise 38. Give a rigorous proof of (1.44), i.e., prove that, for any λ > 0,
Z ∞
(sin(λx))2
dx = π.
λx2
−∞
Exercise 39. * Prove the following variant of Theorem 1.14: Let a(n) be sequence
of complex numbers, b(n) be a sequence of non-negative real numbers, and define
A(s) =
∞
X
a(n)
n=1
ns
and
B(s) =
∞
X
b(n)
n=1
ns
.
Suppose that
• |a(n)| ≤ b(n),
• the series defining B(s) converges for σ > 1, and
• A(s) has an analytic continuation to σ ≥ 1 except for at most a simple
pole at s = 1 with residue r.
Then, as x → ∞,
X
a(n) = rx + o(x).
n≤x
Hint: first, show that without loss of generality you can assume that the a(n) are
all real; for this aim introduce another Dirichlet series having Dirichlet series coefficients a(n) and write A(s) as the sum of two Dirichlet series, one having real
coefficients and one with coefficients in iR.
Exercise 40. Show that any Dirichlet L-function L(s, χ) is non-vanishing on the
line σ = 1.
Hint: consider the function L(σ, χ0 )17 |L(σ + it, χ)|24 |L(σ + 2it, χ2 )|8 and argue as
in the case of the zeta-function.
Chapter 1
48
Classical L-functions
Exercise 41. Assume that f (s) is an analytic function in σ > 1 without zeros (so
that we can define a logarithm) and
log f (s) =
∞
X
a(n)
n=1
ns
with a(n) ≥ 0. Further, suppose that f (s) is analytic on σ = 1 except for a pole of
order m ≥ 0 at s = 1. Prove that if f (s) has a zero s = 1 + it0 , then its order is
≤ m/2.
Hint: If s = 1 + it0 is a zero of order k > m/2, then consider the function
f (s)2k+1
2k
Y
f (s + ijt0 )2(2k+1−j) .
j=1
Exercise 42. * i) Prove the decomposition (1.47).
ii) Verify functional equation (1.49) in the case of quadratic number fields.
√
Exercise 43. * Show that both Q(i) and Q( 5) have class number h = 1.
Hint: use the previous exercise and compute the corresponding value L(1, χ d ).
Let r(n) count the number of ways the positive integer n can be written as a
sum of two integer squares (with repetition, i.e, r(n) = {(a, b) ∈ Z2 : n = a2 + b2 }.
Exercise 44. i) Show that
ζQ(i) (s) =
and deduce that
X
n≤x
∞
1 X r(n)
4 n=1 ns
r(n) ∼ πx.
Hint: use the last but one exercise and Theorem 1.14.
ii) Use geometric arguments in order to prove the last statement with an error term
√
O( x).
√
Hint: how many integer lattice points lie in a circle of radius r = x centered at
the origin?
The circle problem is to find the best possible error term in the mean-value formula
P
for r(n). It is known that | n≤x r(n) − πx| is for infinitely many values of x larger
1
131
than x 4 and always bounded by x 416 +ǫ . The first result is due to Hardy [76] and
Landau [125] (independently); a slight but remarkable improvement of the lower
bound by some log-powers was found by Soundararajan [185]. The upper bound
is from Huxley [93] and there is hope that refinements of techniques in the theory
of exponential sums will lead to a smaller exponent.
Exercise 45. Deduce from the prime ideal theorem for Q(i) and the splitting of
primes that the prime numbers are equidistributed in the prime residue classes
modulo 4.
Section 1.6
The explicit formula
49
1.6. The explicit formula
Now we want to prove Riemann’s explicit formula (1.14) which links the
prime numbers directly with the zeros of the zeta-function. However, while
Riemann dealt with the prime counting function π(x) (see (1.14)) we shall
P
work with the more simple function ψ(x) = pk ≤x log p, introduced in Section 1.4, and prove that, for x 6= pk ,
X xρ 1
1
− log 1 − 2 − log(2π).
(1.51)
ψ(x) = x −
ρ
2
x
ρ
Notice that the right hand side above is not absolutely convergent. If ζ(s)
would have only finitely many nontrivial zeros, the right hand side would be
a continuous function of x, contradicting the jumps of ψ(x) for prime powers
x. The derivation of the explicit formula relies on a more detailed study of
basic analytic properties of the Riemann zeta-function and it provides us a
better understanding on the nature of the error term in the prime number
theorem. Here we shall work in the more general setting of Dirichlet Lfunction; nevertheless, we follow closely von Mangoldt’s original approach
[141] for zeta. First of all we have to recall some facts from the theory of
functions.
1.6.1. Entire functions of finite order. The theory of entire functions
was founded by Weierstrass [208] in 1876 and was further developed in the
1890s by the path-breaking works of Picard and Hadamard [72]. We start
with some observations concerning the zeros of entire functions. The main
tool is Jensen’s formula:
Lemma 1.17. Let f (s) be an analytic function for |s| ≤ r with zeros
ρ1 , . . . , ρm (according their multiplicities) in |s| < r, f (s) 6= 0 for |s| = r,
and f (0) 6= 0. Then
Z 2π
r m |f (0)|
1
log |f (r exp(iθ))| dθ = log
.
2π 0
|ρ1 · . . . · ρm |
Proof. First we assume that f (s) does not vanish for |s| ≤ r; then Jensen’s
formula is an easy consequence of Cauchy’s theorem applied to log f (s) (more
precisely, it is the real part of the resulting formula).
Now assume that f (s) has zeros inside the circle |s| = r. Then we first
consider for any such zero s = ρ the function gρ (s) = s − ρ. Define
G(s) =
It is easily seen that
1
2π
Z
0
gρ (s)
.
− sρ
r2
2π
log |G(r exp(iθ))| dθ = − log r,
Chapter 1
50
Classical L-functions
and
Z
0
2π
log |G(r exp(iθ))| dθ =
Z
0
2π
log |gρ (r exp(iθ))| dθ − 2π log r.
Hence,
1
2π
Z
2π
0
log |gρ(r exp(iθ))| dθ = log r = log |gρ(0)| + log
r
.
|ρ|
Now write f (s) = F (s)(s − ρ1 ) · . . . · (s − ρm ) with non-vanishing F (s) and
apply the already proven parts. Adding all resulting formulas together, yields
Jensen’s formula. •
An entire function f (s) is said to be of finite order if there is a non-negative
real number λ such that
f (s) ≪ exp |s|λ
as |s| → ∞. The infimum over all numbers λ for which this estimate holds
is called the order of f . By Liouville’s theorem, the functions of order zero
are the polynomials.
Our next aim is to show that the zeros of an entire function f of finite order
cannot lie too dense; in fact, their location is related to the order of f .
Theorem 1.18. Let f be an entire function of finite order λ with zeros
ρ1 , ρ2 , . . . arranged so that |ρ1 | ≤ |ρ2 | ≤ . . . and repeated according their
multiplicities. Then
♯{j : |ρj | ≤ r} ≪ r λ+ǫ .
If κ > λ, then
X
ρj 6=0
|ρj |−κ < ∞.
Proof. Without loss of generality we may suppose that f (0) 6= 0. Further
we assume that f (s) does not vanish for |s| = 3r (since the zeros of an entire
function form a discrete set this choice is indeed possible). Since log 3 > 1,
we deduce from Jensen’s formula 1.17 that
X
X
3r
1 ≤
log
|ρj |
|ρj |≤r
|ρj |≤r
Z 2π
1
= − log |f (0)| +
log |f (3r exp(iθ))| dθ
2π 0
≪ r λ+ǫ .
The convergence of the series is a simple consequence. This proves the theorem. •
Section 1.6
The explicit formula
51
1.6.2. Hadamard products. Weierstrass proved that any non-zero entire function can be factored into a product over its zeros (times an exponential function). In the case of polynomials this is just another formulation
of the fundamental theorem of algebra (that any polynomial over C has a
root in C) and is known since Gauss’ first proof in his doctorate. However,
a generic entire function has infinitely many zeros and hence its so-called
Weierstrass product is infinite and the analysis much more difficult. As part
of his theory of entire functions, Hadamard [72] obtained for entire functions
of finite order a more explicit form for Weierstrass’ products. For our purpose
it suffices to consider only functions of order one.
Theorem 1.19. Let f (s) be an entire function of order 1 with zeros ρ0 = 0
with multiplicity m0 and ρ1 , ρ2 , . . . arranged so that 0 < |ρ1 | ≤ |ρ2 | ≤ . . . and
repeated according their multiplicities. Then there are constants A, B such
that
∞ Y
s
s
m0
exp
.
f (s) = s exp(A + Bs)
1−
ρj
ρj
j=1
A proof of this theorem can be found in any textbook on the theory of
functions, e.g., Titchmarsh [199]. Therefore, we shall here give only a sketch
of
Proof. Without loss of generality we may assume that f (0) 6= 0. Since
(1 − z) exp(z) = 1 − z 2 + higher terms,
the product
∞ Y
s
s
exp
P (s) :=
1−
ρj
ρj
j=1
converges absolutely for any s, and so it represents an entire function. Writing
f (s) = P (s)G(s), it follows that G(s) is an entire function without zeros. Now
assume that G is of finite order. We shall show that then G(s) = exp(g(s)),
where g(s) is a polynomial of degree less than or equal to the order λ of G.
To see this consider the entire function h(s) := log G(s) − log G(0). We
write s = r exp(iφ) with φ ∈ R and observe that
Re h(s) = log |G(s)| ≪ r λ+ǫ .
There are real numbers an , bn such that
∞
X
h(s) =
(an + ibn )sn ,
n=0
and therefore
Re h(s) =
∞
X
n=0
(an r n cos(nφ) − bn r n sin(nφ)) .
Chapter 1
52
Classical L-functions
Hence, by Fourier theory,
n
|an |r ≪
Z
0
2π
|Re h(r exp(iφ))| dφ.
It is easily seen that a0 = 0 and
Z 2π
Re h(r exp(iφ)) dφ = 0.
0
Thus,
n
|an |r ≪
Z
0
2π
{|Re h(r exp(iφ))| + Re h(r exp(iφ))} dφ ≪ r λ+ǫ
since |x| + x is equal to 2x if x is positive or equal to zero otherwise. Sending
r → ∞ implies an = 0 for n > 1. This proves the claim on the function G.
We remark that the same argument can be applied if we would know
G(s) ≪ exp(rjλ+ǫ )
(1.52)
for a sequence rj tending to infinity.
It remains to show that G is of order one. In view of the just given remark
we have to verify an estimate of the form (1.52) with λ = 1. For this purpose
we choose rj such that
|rj − |ρn || > |ρn |−2 ;
this choice can be realized since the measure of all intervals (|ρn |−|ρn |−2 , |ρn |+
|ρn |−2 ) is bounded by
∞
X
2
|ρn |−2 ,
n=1
which is finite by Theorem (since f (s) has order one). Now we write P =
P1 P2 P3 , where the Pk = Pk (s) are those parts of the product P (z) according
to
• in P1 : |ρn | < 21 rj ,
• in P2 : 12 rj ≤ |ρn | ≤ 2rj
• in P3 : 2ri < |ρn |.
For the factors in P1 we observe
r
s
s
j
> exp −
1−
exp
.
ρn
ρn |ρn |
Taking into account Theorem 1.18,
∞
r ǫ X
X
j
−1
|ρn |−1−ǫ .
|ρn | ≤
2
1
n=1
|ρn |< 2 rj
Thus it follows that |P1 (s)| > exp(−rj1+ǫ ).
For any factor in P2 we find the lower bound ≫ rj−3 . Since n(rj ) ≪ rj1+ǫ ,
it follows that P2 (s) ≫ exp(−c2 rj1+ǫ ) for some positive constant c2 .
Section 1.6
The explicit formula
53
Finally, for any factor in P3 we get the lower bound
r2
1 − s exp s > exp −c3 j
ρn
ρn |ρn |2
for some positive constant c2 . Similar as in the case of P1 we get |P3 (s)| >
exp(−rj1+ǫ ).
Collecting all estimates for the Pk ’s together, we deduce that
|G(s)| < exp(rj1+ǫ ).
Now it is not difficult to see that G is of the form G(z) = exp(g(z)) with a
polynomial g of degree at most one. This proves the theorem. •
1.6.3. Applications. Now we shall apply the results of the previous
subsections to the zeta-function and to Dirichlet L-function to primitive characters. We start with zeta and define
s
1
− 2s
ζ(s).
ξ(s) = s(s − 1)π Γ
2
2
Notice that here we have removed the simple pole of the zeta-function at
s = 1; the factor s is included with respect to the symmetry of the functional
equation (1.9). We further observe that the zeros of ξ(s) are exactly the
nontrivial zeros of the zeta-fucntion (by the presence of the Gamma-factor
of the functional equation ξ(s) has a non-zero limit for s → −2n). From the
functional equation (1.9) it follows that ξ(s) is an entire function satisfying
ξ(s) = ξ(1 − s).
Recall Stirling’s formula
1
log z − z +
log Γ(z) =
z−
2
1
log z − z +
(1.53)
=
z−
2
Z ∞
[u] − u +
1
log 2π +
2
u+z
0
1
log 2π + O |z|−1 ,
2
1
2
du
the latter asymptotic formula being valid uniformly in z with −π + ǫ ≤
arg z ≤ π − ǫ. Furthermore, we have
(1.54)
(s − 1)ζ(s) ≪ |s|2
for σ ≥ 21 , which follows immediately from Theorem 1.3. This leads to the
estimate
(1.55)
|ξ(s)| < exp(c|s| log |s|)
for some positive constant c as |s| → ∞. Now we consider s → +∞. Since
then ζ(s) → 1, we obtain
s
log s
(1.56)
ξ(s) > exp
4
54
Chapter 1
Classical L-functions
as s → ∞. Taking into account the functional-equation for ξ(s) we obtain
inf λ : |ξ(s)| ≪ exp(|s|λ) = 1.
It thus follows that ξ(s) is an entire function of order 1. So we can apply
Hadamard’s product theorem 1.19 and obtain Riemann’s conjectured product
representation (1.13).
Now we shall show that (1.56) implies that ξ(s) has infinitely many zeros,
resp. the existence of infinitely many nontrivial zeros ρ for ζ(s). For this
purpose assume that
X
(1.57)
|ρ|−1
ρ
is convergent. For any complex number z,
|(1 − z) exp(z)| < exp(2|z|).
Applying this with z = ρs , and taking into account the convergence of (1.57),
we deduce from Hadamard’s product theorem 1.19 that ξ(s) ≪ exp(C|s|) for
some constant C > 0, as |s| → ∞, in contradiction to (1.56). Thus the series
(1.57) diverges. By Corollary 1.18, it converges if we replace the exponent
−1 by anything smaller. This information on the nontrivial zeros of ζ(s) is
new: we did not make use of the (so far unproved Riemann-von Mangoldt
formula). We collect our observations in
Corollary 1.20. There exist constants A and B such that
Y
s
s
exp
,
ξ(s) = exp(A + Bs)
1−
ρ
ρ
ρ
where the product is taken over the nontrivial zeros of ζ(s). Furthermore, the
P
P
series ρ |ρ|−1 diverges while ρ |ρ|−1−ǫ converges for any positive ǫ.
We may argue analogously for Diriclet L-functions. For any primitive character χ mod q, let
q s+δ
s+δ
2
ξ(s, χ) =
Γ
L(s, χ),
π
2
where δ = 12 (1 − χ(−1)). In a similar manner as above we conclude that
ξ(s, χ) is an entire function of order 1 and also the other observations hold
in this case with the difference that the functional equation takes the form
ξ(1 − s, χ) = ξ(s, χ). Thus
Corollary 1.21. Let χ mod q be a primitive character. There exist constants
Aχ , Bχ such that
Y
s
s
exp
,
1−
ξ(s, χ) = exp(Aχ + Bχ s)
ρχ
ρχ
ρ
χ
Section 1.6
The explicit formula
55
where the product is taken over the nontrivial zeros ρχ of L(s, χ). The series
P
P
−1
diverges while ρχ |ρχ |−1−ǫ converges for any positive ǫ.
ρχ |ρχ |
1.6.4. The logarithmic derivative. In the proof of the prime number
theorem we have already worked with the logarithmic derivative. We note
that any zero of a meromorphic function is a simple pole of its logarithmic
derivative, independent of its multiplicity (this follows immediately from the
Laurent expansion). Our next aim is to deduce the partial fraction decomposition of the Riemann zeta-function and Dirichlet L-functions, respectively.
We start again with zeta. Recall that we denote the nontrivial zeros by
ρ = β + iγ.
Theorem 1.22. We have

if
 O(1)
′
P
ζ
1
+ O(1 + log |s|) if
(s) =
 |t−γ|<1 s−ρ
ζ
O(log |s|)
if
σ ≥ 2,
− 1 ≤ σ ≤ 2, |t| ≥ 1,
σ ≤ −1, |s + 2n| > 41 , n ∈ N.
For the proof of this theorem we shall use two results which we have not
proved so far: first, the functional equation (1.9) (which we will prove in
Chapter 2) and, second, a weak form of the Riemann-von Mangoldt formula
(1.12) (which is one of our aims in Chapter 3).
Proof. In the half-plane σ ≥ 2 we find
′ ∞
X
ζ
log n
(s) ≤
,
ζ
2
n
n=2
which leads to the estimate of the theorem.
In the region σ ≤ −1 we use the functional equation; it is not difficult to
see that (1.9) can be rewritten as
πs
(1.58)
ζ(1 − s) = 21−s π −s cos Γ(s)ζ(s).
2
Logarithmic differentiation leads to
ζ′
π
πs Γ′
ζ′
− (1 − s) = − log 2π − tan
+ (s) + (s).
ζ
2
2
Γ
ζ
Now the estimate in question follows from the bound for σ ≥ 2 and Stirling’s
formula (1.53).
In order to obtain the estimate for the region covering the critical strip
we need an easy consequence of the Riemann-von Mangoldt formula (1.12),
namely that
(1.59)
(1.60)
N(T + 1) − N(T − 1) ≪ log T ;
in fact, this can also be deduced directly from Jensen’s formula (which we
leave as an exercise to the interested reader).
Chapter 1
56
Classical L-functions
Now we continue with the final estimate for the region −1 ≤ σ ≤ 2. By
symmetry we may assume that t > 1. Differentiation of the Hadamard
product representation from Corollary 1.20 leads to
X 1
1
ξ′
.
(s) = B +
+
(1.61)
ξ
s
−
ρ
ρ
ρ
Moreover, we have
ξ′
1
1
1
1 Γ′ s ζ ′
+ (s).
(s) = +
− log 2π +
ξ
s s−1 2
2Γ 2
ζ
By Stirling’s formula (1.53),
Γ′
(s) = log s + O
Γ
1
|s|
as |s| → ∞ and −π + ǫ < arg s < π − ǫ. Thus we get
X 1
ζ′
1
+ O(log t).
(s) =
+
ζ
s−ρ ρ
ρ
Using this formula with s = 2 + it and subtracting the resulting formula from
the previous one, we arrive at
X
1
1
ζ′
+ O(log t).
(σ + it) =
−
(1.62)
ζ
σ + it − ρ 2 + it − ρ
ρ
For the first we consider only the terms of the (2 + it−ρ)−1 with |t−γ| ≤ 1.
Each of these terms is bounded and by (1.60) there exist O(log t) many of
them. Hence
X
1
≪ log t.
|2 + it − ρ|
|t−γ|<1
Next we investigate the contribution of the terms with |t − γ| ≥ 1. We have
1
1
2−σ
−
=
≪ (t − γ)−2 .
σ + it − ρ 2 + it − ρ
(σ + it − ρ)(2 + it − ρ)
Again with (1.60) we get
X 1
1
−
σ + it − ρ 2 + it − ρ
|t−γ|≥1
X
X
1
1
=
−
σ
+
it
−
ρ
2 + it − ρ
t+m≤γ≤t+m+1
m∈Z,m6=−1,0
≪
log |t + m|
≪ log t.
2
m
m∈Z,m6=−1,0
X
Substituting this and the previous estimate in (1.62) leads to the formula of
the theorem. The theorem is proved. •
Section 1.6
The explicit formula
57
The same method can be applied to Dirichlet L-functions to primitive characters. Some things are more simple here; e.g., there is no pole at s = 1.
However, other parts need special attention. The trivial zeros are located at
s = δ −2n, n ∈ N0 (where δ = 1 if χ(−1) = −1, and δ = 0 otherwise). In fact,
if χ(−1) = 1, then L(s, χ) has a trivial zero at s = 0 which has to be treated
in a similar manner as the pole of ζ(s). Moreover, we have to make use of
the corresponding functional equation (1.25) for Dirichlet L-functions and
the analogue to the corresponding weak version of the Riemann-von Mangoldt formula. If χ mod q is a primitive character and N(T ; χ) counts the
number of nontrivial zeros ρχ = βχ + iγχ of L(s, χ) with |γχ | ≤ T (according
multiplicities), then
qT
T
log
+ O(log qT );
π
2πe
note that here also zeros from the lower half-plane are counted in lack of a
symmetry with respect to the real axis in case of non-real characters. For the
argument below the following consequence of (1.63) is sufficient:
(1.63)
N(T ; χ) =
N(T + 1; χ) − N(T ; χ) ≪ log T,
where the implicit constant depends on q.
Then the analogue of Theorem 1.23 for Dirichlet L-functions takes the form:
Theorem 1.23. Let χ mod q be a primitive character. We have


O(1)
if σ ≥ 2,


 P
1
′
+
O(log(q(1
+
|s|)))
if − 12 ≤ σ ≤ 2,
L
|t−γχ |<1 s−ρχ
(s, χ) =

L
O(log(q|s|))
if σ ≤ − 12 , and



|s + 2n − δ| > 14 , n ∈ N,
where in the case of χ(−1) = 1 the second estimate holds only for |s| > 12 ;
for |s| ≤ 21 , we have in this case
X
L′
1
1
(s, χ) − =
+ O(log q).
L
s
s − ρχ
|t−γχ |<1
In view of applications we have here incorporated the dependency of the
error term on the character. For this aim we have to be a bit careful: for
example, we cannot bound Bχ by an absolute constant as in the case of the
zeta-function: B ≪ 1. We leave the details to the interested reader.
1.6.5. Proof of the explicit formula. Now we are going to prove another of Riemann’s conjectures, the explicit formula. We prefer to work with
ψ(x), resp. the slightly modified function
ψ(x)
if x 6∈ Z,
ψ0 (x) = P
1
n<x Λ(n) + 2 Λ(x) if x ∈ Z.
58
Chapter 1
Classical L-functions
This modification is made with respect to obtain an exact formula also in
the case of integral x. As a matter of fact, we can replace Perron’s formula
(1.36) by
Z c+i∞ ′
xs
ζ
1
(1.64)
(s) ds.
ψ0 (x) = −
2πi c−i∞ ζ
s
We observe that this formula is valid for any x ∈ R since we have added the
term 21 Λ(x) for x being an integer (with respect to the contribution of the
term for y = nx = 1 in Lemma 1.12).
Moving now the path of integration to the left, we find that the latter
expression is equal to the corresponding sum of residues, that are the residues
of the integrand at the pole of ζ(s) at s = 1, at the zeros of ζ(s), and at the
pole of the integrand at s = 0. We have already identified the main term as
being the residue at s = 1. Each zero ρ gives the contribution
′
xρ
ζ
xs
= .
Res s=ρ
(s)
ζ
s
ρ
In particular, for the trivial zeros ρ = −2n with n ∈ N we get the contribution
∞
X
1
1
x−2n
= − log 1 − 2 .
−2n
2
x
n=1
The simple pole at s = 0 leads to
′
ζ
ζ ′ xs
xs
ζ′
(1.65)
Res s=0
= lim s (s) = (0) = log(2π)
(s)
s→0 ζ
ζ
s
s
ζ
(the computation of this constant is left to the reader as an easy exercise).
This leads to the exact explicit formula (1.51)
X xρ 1
1
− log 1 − 2 − log(2π),
(1.66)
ψ0 (x) = x −
ρ
2
x
ρ
being valid for any positive real x, which is slightly stronger than (1.51) and
equivalent to Riemann’s version (1.14). However, for a rigorous proof we
have to prove that the integrals over the contour vanish. We include this
in a study of a truncated version of the explicit formula. Here we shall cut
the integral (1.64) at t = ±T ; of course, in this setting we will have error
terms, but the resulting version of the explicit formula is rather convenient
for applications.
Theorem 1.24. For x > 2,
X xρ 1
1
− log 1 − 2 − log(2π) + R(x, T )
ψ0 (x) = x −
ρ
2
x
|γ|≤T
with
R(x, T ) ≪
x
(log(xT ))2 + min{1, x/(T hxi)} log x,
T
Section 1.6
The explicit formula
59
where hxi denotes the minimum of |x − pk | for x 6= pk where p is prime and
k ∈ N.
Notice that for x = pk we have hxi = 0 and the minimum appearing in the
error term takes the value 1. Furthermore, we observe that R(x, T ) vanishes
as T → ∞ and thus the theorem implies (1.66). The convergence will turn
out to be uniform in any closed interval which does not contain a prime power
(for which ψ0 (x) is discontinuous).
Proof. We put c = 1+ log1 x in (1.37); then xc = ex. Since c is a function of x,
we have to be a bit more careful than in the proof of Theorem 1.9; however,
much of the reasoning follows just the same way. First of all we find
Z c+iT ′
xs
ζ
1
(s) ds
ψ0 (x) +
2πi c−iT ζ
s
X
x c
Λ([x])
Λ(n)
,
min{1, (T | log x/n|)−1 } + c
≪
n
T
x6=n∈N
where the last term can be omitted if x 6∈ N. Similar as before we find that
x
). For n ∈ ( 12 x, 2x)
the contribution of the terms with n 6∈ [ 21 x, 2x] is O( x log
T
we define x1 to be the maximal prime power pk < x. If x1 ≤ 12 x, then
Λ(n) = 0 for all n under consideration and we are done. If x1 > 21 x, we
consider the term with n = x1 separately. Since then
log
x − x1
hxi
x
≥
≥
,
x1
x
x
we find
x c
min{1, (T | log x/x1 |)−1 } ≪ min{1, x/(T hxi)} log x.
n
The other terms can be estimated as before. Hence we obtain
Z c+iT ′
ζ
1
xs
ψ0 (x) = −
(s) ds +
2πi c−iT ζ
s
x(log x)2
(1.67)
+ min{1, x/(T hxi)} log x .
+O
T
Λ(n)
In the next step we apply the calculus of residues to the contour integral
taken over the rectangular path with corners −U ± iT and c ± iT , where U
is a positive odd integer (to have some distance from the trivial zeros of zeta
which are simple poles of the integrand). Here we may estimate the integral
over the vertical segment just as we did before; we obtain
Z T ′
Z −U +iT ′
ζ
xs
log U
x−U
ζ
(−U + it)
(1.68)
(s) ds ≪
dt ≪
T.
ζ
s
ζ
| − U + it|
UxU
−U −iT
−T
However, the vertical integrals need special attention since here the segments
may run through a neighbourhood of a trivial zero. In view of (1.60) (resp.
Chapter 1
60
Classical L-functions
the Riemann-von Mangoldt formula) there are at most O(log n) trivial zeros
ρ = β + iγ with n < γ ≤ n + 1. Hence there exists a T = Tn ∈ (n, n + 1] such
that
1
(1.69)
|T − γ| ≫
log T
for all zeros ρ. For s = σ + iT we deduce from Theorem 1.22 and once more
(1.60) that
X
ζ′
1
(s) =
+ O(log T ) ≪ (log T )2 .
ζ
s−ρ
|T −γ|<1
This estimate together with the corresponding one of Theorem 1.60 for σ ≤
−1 leads to
Z −1 s Z c s
Z c±iT ′
x x ζ
xs
2
log |s| dσ
(log
T
)
dσ
+
(s) ds ≪
s
s
ζ
s
−U
−U ±iT
−1
x (log T )2
.
≪
log x T
One observes that all these estimates are uniform in U. Moreover we note
that (1.68) tends to zero as U → ∞. Hence we arrive at the explicit formula
as given in the theorem, valid for those T which satisfy (1.69); however, the
latter condition can be relaxed. By construction, for any T > 1 there is a
Tn which has distance less than 1 from T and satisfies (1.69). Obviously,
substituting T has no influence on the size of the error term R(x, T ) and for
the sum over the nontrivial zeros we observe that
X xρ
x log T
≪
ρ
T
|γ|∈[T,Tn ]
by (1.60). This can be absorbed in R(x, T ). This proves the theorem. •
In a rather similar way one can prove explicit formulae for Dirichlet Lfunctions. If χ mod q is a primitive character and ρχ denotes the nontrivial
zeros of L(s, χ), then we find analogously
ψ0 (x, χ) = −
if χ(−1) = −1, and
ψ0 (x, χ) = −
X xρχ
ρχ
ρχ
X xρχ
ρχ
ρχ
∞
X
L′
x1−2n
− (0, χ) +
L
2n − 1
n=1
− log x − lim
s→0
L′
1
(s, χ) −
L
s
+
∞
X
x−2n
n=1
2n
otherwise, i.e., if χ(−1) = +1; here ψ0 (x, χ) is the function ψ(x, χ) =
P
n≤x Λ(n)χ(n) modified in just the same way as we did when we switched
from ψ(x) to ψ0 (x). The origin of most of the appearing terms is clear; however, we should have a brief look on the main difference. The logarithmic
Section 1.6
The explicit formula
61
derivative of L(s, χ) at s = 0 is regular if χ(−1) = −1 and it has a simple
pole if χ(−1) = +1 (by the trivial zero of L(s, χ)). As in the proof of Theorem 1.23, these cases have to be considered separately and the result are the
slightly differing explicit formulae above.
Again it is desirable to have truncated versions which are uniform in the
modulus q. This is a rather difficult task, in particular, since we cannot
exclude nontrivial zeros ρχ near s = 0 or 1. One can show that such a socalled exceptional (or Siegel-) zeros can only occur if χ is a real character and
it is itself real. One can also show that there cannot be too many of these
zeros; however, we do not want to go into the details and simply state the
following result without proof:
Theorem 1.25. Let χ mod q be a non-principal character and assume that
2 < T ≤ x. Then
′
X xρχ
xβχ
+ R(x, T, q),
ψ(x, χ) = − ′ −
βχ
ρχ
|γχ |≤T
where
R(x, T, q) ≪
1
x
(log(qx))2 + x 4 log x.
T
′
βχ
The term xβ ′ is to be omitted unless χ is a real character for which L(s, χ)
χ
vanishes at s = βχ′ satisfying the estimate
βχ′ > 1 −
c
,
log q
where c is a positive absolute; if the zero βχ′ exists, the sum has to be taken
over all nontrivial zeros in the given range different from βχ′ and 1 − βχ′ .
Here we have also included the case of non-primitive characters and ψ0 (x, χ)
is replaced by ψ(x, χ) which is more useful for applications. For a proof we
refer once more to Davenport [44] and Prachar [172].
1.6.6. Improvement on the error term in the prime number theorem. The explicit formula allows a remarkable improvement on the error
term in the prime number theorem. In the sequel we focus on the Riemann
zeta-function; for the more general case of Dirichlet L-functions (where one
wants to have uniformity in the module) we refer to Prachar [172].
First of all, we observe that our deeper knowledge on the analytic behaviour
of the zeta-function implies a larger zero-free region inside the critical strip.
Lemma 1.26. There exists a constant c > 0 such that for any nontrivial
zero ρ = β + iγ
β < 1 − C min{1, (log |γ|)−1 }.
Chapter 1
62
Classical L-functions
Proof. We shall use the same ideas as in the proof of Lemma 1.11 but now
we incorporate the approximation for ζ(s) from Theorem 1.22, i.e.,
X
1
ζ′
(s) =
+ O(log |s|).
ζ
s−ρ
|t−γ|<1
For the first we suppose that σ > 1 and without loss of generality let t > 0.
We observe that the real part of the summands is positive for σ > 1. Hence,
we deduce that
′ 1
ζ
+ c log |s|
Re − (s) < −Re
ζ
s−ρ
for any nontrivial zero ρ = β + iγ with |t − γ| < 1, where c > 0 is a suitable
constant; of course, here we can also delete the ρ-term on the right (since its
contribution is negative) and obtain
′ ζ
Re − (s) < c log |s|.
ζ
Now recall (1.30). Using the latter estimate with t = γ it follows (in just the
same way as in the proofs of Lemma 1.10 and 1.11) that
0≤
resp.
24
17
−
+ c log(t + 2),
σ−1 σ−ρ
β <1+
by putting σ = 1 +
δ
log(t+2)
4δ
1
−
log(t + 2) (3 + cδ) log(t + 2)
with δ > 0. This proves the lemma. •
We may use the just proved lemma in order to obtain the following improvement on the prime number theorem 1.9:
Theorem 1.27. There exists an absolute positive constant C such that for
sufficiently large x
1
.
π(x) = Li (x) + O x exp −C(log x) 2
This is for many application indeed a valuable improvement, however, it is
still weaker than (1.39) which can be obtained by incorporating the so far
best zero-free region for the zeta-function.
P ρ
Proof. We consider the sum ρ xρ appearing in the explicit formula. For
each term with |γ| ≤ T we find by the previous lemma that
log x
ρ
.
x ≪ x exp −C
log T
Section 1.6
The explicit formula
63
Furthermore, we have
X 1
X 1
≤
.
|ρ| 0<γ≤T γ
|γ|≤T
To bound this sum we apply partial summation in conjunction with Riemannvon Mangoldt formula (1.12) in the weak form
N(T ) ≪ T log T
(recall that (1.12) was still not proved in these notes but that we will return
to this problem in Chapter 3). Then we find that the sum in question is
Z T
N(T )
N(t)
=
+
dt ≪ (log T )2 .
2
T
t
0
Hence,
X xρ
log x
2
.
≪ x(log T ) exp −C
ρ
log T
|γ|≤T
Without loss of generality we may suppose that x is a positive integer. Then
we deduce from the explicit formula, Theorem 1.24, that
log x
x(log xT )2
2
.
+ x(log T ) exp −C
ψ(x) − x ≪
T
log T
1
Taking the balance (log T )2 = log x and T = exp((log x) 2 ) respectively, we
may deduce the bound of Theorem 1.27 by partial summation. •
1.6.7. Weil’s explicit formula. The explicit formula combines the remarkable fact that the zeta-function (resp. any Dirichlet L-function) can be
written both as an Euler product over the primes and as a Hadamard product
over the trivial zeros. Weil [210] proved a rather general extension of this
reciprocity between primes and zeros. In order to state his result we have to
introduce some new notions.
Let f be a measurable function on R. We say that f is of type (αℓ , αr ),
where αℓ < αr are real numbers, if x 7→ f (x)|x|σ−1 is integrable for σ ∈
(αℓ , αr ). In this case we define the Mellin transform of f by
Z ∞
M(f, s) =
f (x)|x|s−1 dx
−∞
for σ ∈ (αℓ , αr ). Under certain circumstances there is an inverse transform
Z σ+i∞
1
M(f, s)|x|−s ds.
f (x) =
2πi σ−i∞
We have already seen some examples of such pairs of transforms in the proof
of the prime number theorem (and therefore it is not surprising to find them
here once again).
Now Weil’s explicit formula takes the form:
64
Chapter 1
Classical L-functions
Theorem 1.28. Let f be a function of type (αℓ , αr ), where αℓ < − 12 and
αr > 21 . Suppose that there exist c, ǫ > 0 such that
|M(f, s)| < c(1 + |s|)−1−ǫ
for all s ∈ [αℓ , αr ], f is of bounded total variation, and that f (x) = 0 if x < 0.
Define
Z ∞
N
dx
,
− f (1) log
f (x)FN (x)
∆∞ (f ) = lim
N →∞
x
π
0
where

− 1 1−x2N

if 0 < x < 1,
 x 2 |x−x−1 |
FN (x) =
0
if x = 1,

 x+ 12 1−x−2N
if x > 1.
|x−x−1 |
The limit in this definition exists and one has
X
X
1
1
k − |k|
+ M f, −
+
∆∞ (f ) +
log p
f (p )p 2 = M f,
2
2
p
06=k∈Z
X 1
−
M f, ρ −
;
2
ρ
all series are absolutely convergent.
Notice that on the left-hand side the summation is taken over all nonequivalent valuations of Q: the p-adic non-archimedean valuations plus the
archimedean absolute value (indicated by the index ∞). For the rather
lengthy proof we refer to Weil [210] and Patterson [164], respectively.
Exercise 46. Fill the gaps in the proofs of Lemma 1.17 and Theorem 1.18.
Exercise 47. i) Prove the Hadamard product representation for the reciprocal of
the Gamma-function:
∞ s
Y
s
1
1+
exp −
,
= s exp(Cs)
Γ(s)
n
n
n=1
where C is the Euler-Mascheroni constant.
ii) What are the residues?
iii) Derive an analogous formula for the sin-function.
Exercise 48. Prove formula (1.55).
Exercise 49. * i) Show for the constants in Corollary 1.20 that
X
ζ′
C
1
A = − (0) = − log 2
and
B=−
ρ−1 = − − 1 + log 4π,
ζ
2
2
ρ
where C is the Euler-Mascheroni constant and the summation in the B-defining
series is such that the terms ρ and 1 − ρ are added together. Deduce that there are
Section 1.6
The explicit formula
65
no zeros ρ = β + iγ with |γ| ≥ 6.
Hint: functional equation plus (1.61)
ii) Prove (1.65).
Exercise 50. i) Deduce the functional equation for the zeta-function in the form
(1.58) from (1.9).
Hint: use basic facts and identities from the theory of the Gamma-function.
ii) Show that any zero of ζ ′ (s) on the critical line is also a zero of ζ(s)
It is expected that ζ ′ (s) does not vanish on the critical line; more precisely, that
all zeros of ζ(s) are simple. Speiser [186] has shown that the Riemann hypothesis
is equivalent to the non-vanishing of ζ ′ (s) in 0 < σ < 21 ; if also ζ ′ ( 21 + it), then, by
ii), all zeros are simple!
Exercise 51. * i) Verify all sketched estimates in the proof of Theorem 1.22.
ii) Show that, for σ ≥ −5, |t| ≥ 1,
11
|ζ(s)| ≪ |t| 2 .
Hint: use the functional equation and Stirling’s formula.
iii) Prove (1.60) without using the Riemann-von Mangoldt formula and deduce the
estimate N (T ) ≪ T log T .
Hint: use Jensen’s formula together with ii).
iv) Prove Theorem 1.23 along the lines of the proof of Theorem 1.22.
Exercise 52. * Prove Theorem 1.25.
Hint: for inspiration one may have a look into [172].
Exercise 53. * i) Deduce Riemann’s explicit formula (1.14) from Theorem 1.24.
ii) Deduce Theorem 1.24 from Weil’s explicit formula.
CHAPTER 2
Zero-distribution of the Riemann zeta-function
In this chapter we shall have a closer look at the zeros of the Riemann zetafunction inside the critical strip. In view of the unsolved Riemann hypothesis
they are the most important objects but also the most difficult to deal with.
We shall show that there are infinitely many zeros on the critical line and
that there cannot be too many nontrivial zeros off the critical line; here we
mean that the proportion of the set of possible violations of the Riemann
hypothesis is zero. Most of the presented methods can be easily generalized
to other L-functions (e.g., Dirichlet L-functions).
2.1. The Riemann hypothesis
The famous Riemann hypothesis states that all nontrivial zeros lie on the
critical line σ = 21 . We can rewrite this equivalently as
Riemann’s hypothesis. ζ(s) 6= 0 for σ > 12 .
There has been a lot of speculation how Riemann was led to this conjecture. One of the reasons might have been his own computations (which are
preserved among his unpublished manuscripts in the library of Göttingen
University). Clearly, in view of the symmetry dictated by the functional
equation the scenario that all zeros lie on the vertical line passing through
the point of symmetry s = 12 is the most beautiful one. But we will never
know what Riemann’s motivation was.
Many computations were done to find a counterexample to the Riemann
hypothesis. Van de Lune, te Riele & Winter [139] localized the first
1 500 000 001 zeros, all lying without exception on the critical line. Moreover all so far localized nontrivial zeros turned out to be simple! Besides
Riemann’s hypothesis we have the
Essential simplicity hypothesis. All (or at least almost all) zeros of ζ(s)
are simple.
2.1.1. The error term in the prime number theorem. The next
result highlights the intimate relation between the zeros of the zeta-function
and prime number distribution.
66
Section 2.1
The Riemann hypothesis
67
Theorem 2.1. For fixed θ ∈ [ 21 , 1),
ψ(x) − x ≪ xθ+ǫ
⇐⇒
ζ(s) 6= 0
for σ > θ.
Our main tool for its proof is the explicit formula from the previous section
which puts the prime numbers in an explicit relation to the nontrivial zeros
of the zeta-function.
Proof. Recall (1.28). For σ > 1 we have
Z ∞
s
ψ(u) − u
ζ′
+s
du.
− (s) =
ζ
s−1
us+1
1
If ψ(x) − x ≪ xθ+ǫ , then the integral above converges for σ > θ, giving an
analytic continuation for
ζ′
1
(s) −
ζ
s−1
to the half-plane σ > θ, and, in particular, ζ(s) does not vanish there.
Conversely, if all nontrivial zeros ρ = β + iγ satisfy β ≤ θ, then it follows
from the explicit formula, Theorem 1.24, that
X 1
x
(2.1)
+ (log(xT ))2 .
ψ(x) − x ≪ xθ
|γ| T
|γ|≤T
In view of (1.60) we get
[T ]
[T ]+1
X
X 1
X log m
X 1
=2
≪
≪ (log T )2 .
|γ|
γ
m
m=1 m<γ≤m+1
m=1
|γ|≤T
Substituting this in (2.1) leads to
x
(log(xT ))2 .
T
finishes the proof of this implication. •
ψ(x) − x ≪ xθ (log T )2 +
Now the choice T = x1−θ
Taking into account Theorem 2.1, we find by partial summation that
π(x) − Li (x) ≪ xθ+ǫ
⇐⇒
ζ(s) 6= 0 for σ > θ.
Now the impact of the Riemann hypothesis on the prime number distribution
becomes visible. If the Riemann hypothesis is true, we may take θ = 21 in
Theorem 2.1 and the resulting estimate for the error term in the prime number
theorem is
1 ψ(x) = x + O x 2 +ǫ .
A slight stronger bound was first obtained by von Koch [118, 119] under
assumption of the Riemann hypothesis (actually, he replaced xǫ by powers of
log x).
68
Chapter 2
Zero-distribution
With regard to known zeros of ζ(s) on the critical line it turns out that an
error term with θ < 21 is impossible. In fact one can show that
1 log log log x
2
(2.2)
π(x) − Li (x) = Ω± x
log x
(see, e.g., Ingham [97]). We have to explain the Ω-notation: given two
functions f (x) and g(x), where g(x) is positive for sufficiently large x, we
write f (x) = Ω+ (g(x)) (resp. f (x) = Ω− (g(x))) if
|f (xn )| ≥ cg(xn )
(resp. |f (xn )| ≤ −cg(xn ))
holds with a positive constant c for some sequence xn which tends to infinity.
Thus, (2.2) shows that π(x) − Li (x) changes its sign infinitely often and that
an error term O(xθ ) with θ < 12 is impossible. In some sense, Riemann’s
hypothesis states that the prime numbers are as uniformly distributed as
possible!
Maybe one of the most given arguments in favour for the truth of Riemann’s
hypothesis is the function field analogue. Davenport and, in particular, Hasse
proved that the so-called Riemann hypothesis for elliptic curves is true, and
later Weil proved the general case of abelian varieties. It is far beyond the
scope of our notes to give an adequate introduction to this topic (nevertheless,
in the following chapter we will briefly explain the meaning of Hasse’s work on
elliptic curves), especially since the analogue of the zeta-function for abelian
varieties is a rational function and so its value-distribution is a priori rather
different than the one of the transcendental function ζ(s). On the other side,
the parallel world of function fields has often been proved to be a signpost
for challenges.
In the following section we present some further heuristics in favour for the
Riemann hypothesis.
2.1.2. Denjoy’s probabilistic argument for Riemann’s hypothesis. Recall the definition of Möbius’ µ-function: we write µ(1) = 1, µ(n) = 0
if n has a quadratic divisor, and µ(n) = (−1)r if n is the product of r distinct
primes. It is easily seen that µ is multiplicative and appears as coefficients of
the Dirichlet series representation of the reciprocal of the zeta-function: for
σ > 1,
X
∞
Y
1
µ(n)
1
=
1− s =
.
ζ(s)
p
ns
n=1
p
Riemann’s hypothesis is equivalent to
X
1
(2.3)
M(x) :=
µ(n) ≪ x 2 +ǫ .
n≤x
This is related to the estimates of Theorem 2.1 (for a proof see, for example,
Titchmarsh [200], §14.25).
Section 2.1
The Riemann hypothesis
69
Denjoy [46] argued as follows. Assume that {Xn } is a sequence of random
variables with distribution
1
P(Xn = +1) = P(Xn = −1) = .
2
Define
S0 = 0
and
Sn =
n
X
Xj ,
j=1
then {Sn } is a symmetrical random walk in Z2 with starting point at 0. A
simple application of Chebyshev’s inequality yields, for any positive c,
1
P{|Sn | ≥ cn 2 } ≤
1
,
2c2
which shows that large values for Sn are rare events. By the theorem of
Moivre-Laplace this can be made more precise. It follows that
2
Z c
o
n
1
1
x
dx.
lim P |Sn | < cn 2 = √
exp −
n→∞
2
2π −c
Since the right hand side above tends to 1 as c → ∞, we obtain
n
o
1
lim P |Sn | ≪ n 2 +ǫ = 1
n→∞
for every ǫ > 0. If the values of the µ-function would behave like random
variables, then Riemann’s hypothesis would hold with probability one! The
law of the iterated logarithm would even give the stronger estimate
o
n
1
lim P |Sn | ≪ (n log log n) 2 = 1,
n→∞
1
which suggests for M(x) the upper bound (x log log x) 2 . This estimate is
pretty close to the so-called weak Mertens hypothesis which states
2
Z X
M(x)
dx ≪ log X.
x
1
Note that this bound implies the Riemann hypothesis and the essential simplicity hypothesis. On the contrary, Odlyzko & te Riele [162] disproved the
original Mertens hypothesis [145],
1
|M(x)| < x 2 ,
by showing
lim inf
x→∞
M(x)
x
1
2
< −1.009
and
for more details see Titchmarsh [200], §14.
lim sup
x→∞
M(x)
1
x2
> 1.06;
Chapter 2
70
Zero-distribution
2.1.3. Approaches towards RH and a substitute. There are some
interesting recent approaches to be mentioned. The first one is an output
of Connes’ theory of non-commutative geometry. Connes [34] obtained a
so-called trace formula in non-commutative geometry which has remarkable
similarity with Weil’s explicit formula, Theorem 1.28. Assuming the Riemann
hypothesis, he shows that this is indeed the explicit formula in disguise and
so this gives a natural spectral interpretation of the nontrivial zeros. This
approach restored some hope to an old idea of Hilbert and Polya that the
Riemann hypothesis follows from the existence of a self-adjoint Hermitian
operator whose spectrum of eigenvalues corresponds to the set of nontrivial
zeros of the zeta-function.
Another approach is from Bombieri [24]. It is based on Weil’s explicit
formula and his positivity criterion for the Riemann hypothesis. The latter
can be rewritten in terms of the positivity of a certain linear functional; then
it is shown that if the Riemann hypothesis is false, then the extremals, in
various relevant Hilbert spaces, would have distinctly unusual properties.
So far we sketched much of the theory for Dirichlet L-functions; however,
the exceptional zeros make their analysis much more complicated and we do
not want to go further into the details but give a glimpse on the impact of their
zero-distribution on the prime number distribution in arithmetic progressions.
All analogues of Riemann’s hypothesis for the whole class of Dirichlet Lfunctions are summed up in the so-called
Generalized Riemann hypothesis. Neither ζ(s) nor any L(s, χ) has a
zero in the half-plane Re s > 12 .
Under assumption of this conjecture one has
π(x; a mod q) =
(2.4)
1
1
Li (x) + O x 2 log(qx)
ϕ(q)
for x ≥ 2, q ≥ 1, and a coprime with q, the implicit constant being absolute.
As long as we do not have a proof of Riemann’s hypothesis in many instances we are often forced to prove conditional results. However, sometimes
one can also find an appropriate way to circumvent the assumption of RH.
We conclude with a remarkable substitute of the Riemann hypothesis, the
celebrated theorem of Bombieri-Vinogradov due to Bombieri [23] and Vinogradov [205] (independently, with a slightly weaker range for Q):
Theorem 2.2. For any A ≥ 1,
X
max max π(y; a mod q) −
q≤Q
a mod q
(a,q)=1
y≤x
1
x
1
Li(y) ≪
+ Qx 2 (log Qx)6 .
A
ϕ(q)
(log x)
Section 2.1
The Riemann hypothesis
71
This shows that the error term in the prime number theorem for arithmetic
1
progressions is, on average over q ≤ x 2 (log x)−A−7 , of comparable size as
predicted by the Riemann hypothesis (see (2.4)).
Exercise 54. Prove that, for fixed θ ∈ [ 21 , 1),
π(x) − Li (x) ≪ xθ+ǫ
⇐⇒
ζ(s) 6= 0
for
σ > θ.
Show that if the Riemann hypothesis is true, then
1
π(x) − Li (x) ≪ x 2 log x.
Exercise 55. i) Prove that Riemann’s hypothesis is equivalent to the estimate
(2.3).
ii) Verify all probabilistic statements in Section 2.1.2.
1
iii) Show that the estimate M (x) ≪ x 2 implies both the Riemann hypothesis and
that all zeros of ζ(s) are simple.
Hint: Show that then
Z ∞
1
M (x)
=s
dx
ζ(s)
xs+1
1
holds for σ > 12 and deduce the estimate
with some positive constant c.
1
|s|
≤c
|ζ(s)|
σ − 12
The essential simplicity conjecture (almost all zeros of ζ(s) are simple) has arithmetical consequences. Cramér [41] showed, assuming the Riemann hypothesis,
Z X
X m(ρ) 2
1
ψ(x) − x 2
dx ∼
ρ ,
log X 1
x
ρ
where the sum is taken over distinct nontrivial zeros ρ and m(ρ) denotes their
multiplicity. The right-hand side is minimal if all the zeros are simple.
Exercise 56. Give an unconditional estimate for the left-hand side. Prove that
the series on the right-hand side converges.
Exercise 57. * Assuming the generalized Riemann hypothesis, prove the asymptotic formula (2.4).
The following sections of this chapter specialize on the Riemann zetafunction and its zero-distribution; however, many of the results can be generalized to other L-functions, e.g., Dirichlet L-functions (unconditionally) or
Dedekind zeta-functions (at least conditionally); for a more general approach
we refer the interested reader to Iwaniec & Kowalski [101] and Lekkerkerker
[129]. In some places we are rather brief since an adequate presentation of
Chapter 2
72
Zero-distribution
all relevant results would be far beyond these notes. Often we also leave the
stage of the classical theory.
2.2. The approximate functional equation
The aim of this section is to prove an approximation of the zeta-function
inside the critical strip:
Theorem 2.3. We have, uniformly for σ ≥ σ0 > 0, |t| ≤ 4x,
X 1
x1−s
−σ
+
+
O
x
.
ζ(s) =
ns s − 1
n≤x
This approximation is a refinement of (1.11) and will turn out to be a rather
useful tool in later applications.
2.2.1. Euler’s summation formula. Let f (u) be any function with
continuous derivative on the interval [a, b]. By partial summation we get
Z b
X
f (n) = ([b] − [a])f (b) −
([u] − [a])f ′ (u) du
a<n≤b
a
= [b]f (b) − [a]f (a) −
Z
b
[u]f ′ (u) du,
a
where [u] = max{z ∈ Z : z ≤ u}. Obviously,
Z b
Z b
Z b
1
1
′
′
f (u) du −
f ′ (u) du.
u−
−
[u]f (u) du =
u − [u] −
2
2
a
a
a
Applying partial integration to the last integral on the right-hand side, we
deduce Euler’s summation formula:
Lemma 2.4. Assume that f : [a, b] → R has a continuous derivative. Then
Z b
Z b
X
1
f ′ (u) du
f (n) =
f (u) du +
u − [u] −
2
a
a
a<n≤b
1
1
+ a − [a] −
f (a) − b − [b] −
f (b).
2
2
Why is this interesting? Imagine we are interested in describing the divergence of the harmonic series in a quantitative way. In such questions it is
often an advantage to work with integrals rather than sums. An easy application of the previous lemma yields the asymptotic formula (1.4) which
describes very precisely the rate of divergence of the harmonic series. However, we are heading for something more difficult. For this purpose we first
replace in Euler’s summation formula the function u − [u] − 21 by its Fourier
series expansion.
Section 2.2
The approximate functional equation
73
Lemma 2.5. For u ∈ R \ Z,
X
1
exp(−2πimu)
1
,
≤
u − −
2πM(u − [u])
2 |m|≤M
2πim
m6=0
and, for u ∈ R,
∞
X
exp(−2πimu)
u − [u] −
=
0
2πim
m6=0
1
2
if u 6∈ Z,
if u ∈ Z,
where the terms with ±m have to be added together; the partial sums are
uniformly bounded in u and M.
Proof. By symmetry and periodicity it suffices to consider only the case
0 < u ≤ 21 . Since
Z 1
2
(−1)m+1 + exp(−2πimu)
exp(−2πimx) dx =
2πim
u
for 0 6= m ∈ Z, we obtain
Z 1 X
X exp(−2πimu)
2
1
exp(2πimx) dx
−u+
=
2πim
2
u
|m|≤M
|m|≤M
m6=0
(2.5)
=
Z
u
1
2
sin((2M + 1)πx)
dx.
sin(πx)
By the mean-value theorem there exists ξ ∈ (u, 21 ) such that the latter integral
is equal to
Z ξ
sin((2M + 1)πx)
dx.
sin(πu)
u
This immediately implies both formulas of the lemma. It remains to show
that the partial sums of the Fourier series are uniformly bounded in u and
M. Substituting y = (2M + 1)πx in (2.5), we get
Z 1
Z 1
2 sin((2M + 1)πx)
2 sin((2M + 1)πx)
dx =
dx +
sin(πx)
πx
u
u
Z 1
2
1
1
sin((2M + 1)πx)
dx
−
+
sin(πx) πx
u
Z 1
Z ∞
2 1 sin(y)
1
dy +
≪
sin(πx) − πx dx
y
0
0
with an implicit constant not depending on u and M; obviously both integrals
exist, which gives the uniform boundedness. •
Chapter 2
74
Zero-distribution
2.2.2. Van der Corput’s summation formula. In 1921, van der Corput [40] invented a new and rather efficient technique to estimate exponential
sums.
Theorem 2.6. For any given η > 0, there exists a positive constant C =
C(η), depending only on η, with the following property: assume that f :
[a, b] → R is a function with continuous derivative, g : [a, b] → [0, ∞) is a
differentiable function, and that f ′ , g and |g ′| are all monotonically decreasing.
Then
X
g(n) exp(2πif (n))
a<n≤b
X
=
f ′ (a)−η<m<f ′ (b)+η
Z
a
b
g(u) exp(2πi(f (u) − mu)) du + E,
where
|E| ≤ C(η) (|g ′(a) + g(a) log(|f ′ (a)| + |f ′ (b)| + 2)) .
Van der Corput’s summation formula looks very technical but the underlying
idea is rather simple. The integral
Z
a
b
g(u) exp(2πi(f (u) − mu)) du
is (up to a constant factor) the Fourier transform of g(u) exp(2πif (u)) at
u = m. Therefore, one can interpret Theorem 2.6 as an approximate version
of Poisson’s summation formula (a topic we will return to in the following
chapter).
Before we can give the proof we shall give the following estimate for exponential integrals.
Lemma 2.7. Assume that F : [a, b] → R has a continuous non-vanishing
derivative and that G : [a, b] → R is continuous. If FG′ is monotonic on [a, b],
then
Z b
G G G(u) exp(iF (u)) du ≤ 4 ′ (a) + 4 ′ (b) .
F
F
a
Proof. First, we assume that F ′ (u) > 0 for a ≤ u ≤ b. Since (F −1 (v))′ =
F ′ (F −1 (v))−1, substituting u = F −1 (v) leads to
Z
b
G(u) exp(iF (u)) du =
a
Z
F (b)
F (a)
G(F −1 (v))
exp(iv) dv.
F ′ (F −1 (v))
Section 2.2
The approximate functional equation
75
Application of the mean-value theorem gives, in case of a monotonically increasing FG′ ,
)
G(F −1 (v))
Re
exp(iv) dv
′
−1 (v))
F (a) F (F
Z ξ
Z F (b)
G
G
(F (a))
cos v dv + ′ (F (b))
cos v dv
=
F′
F
F (a)
ξ
(Z
F (b)
with some ξ ∈ (a, b). The same argument applies to the imaginary part. The
case F ′ (u) < 0 can be treated analogously. This gives the desired estimate.
The lemma is proved. •
Now we are in the position to give the
Proof of Theorem 2.6. Using Euler’s summation formula with F (u) =
g(u) exp(2πif (u)) and the Fourier series expansion of Lemma 2.5, we get
X
g(n) exp(2πif (n))
a<n≤b
=
Z
b
g(u) exp(2πif (u)) du + O(g(a))
a
Z bX
exp(−2πimu) d
(g(u) exp(2πif (u))) du.
+
2πim
du
a m6=0
Since the series on the right-hand side converges uniformly on each compact
subset which is free of integers, and since its partial sums are uniformly
bounded, we may interchange summation and integration. This yields
X
g(n) exp(2πif (n)) =
a<n≤b
Z
b
g(u) exp(2πif (u)) du
a
X 1 1
+
I1 (m) +
I2 (m) + O(g(a),
m
2πi
m6=0
where
I1 (m) :=
I2 (m) :=
Z
Z
b
f ′ (u)g(u) exp(2πi(f (u) − mu)) du,
a
b
a
g ′(u) exp(2πi(f (u) − mu)) du.
Chapter 2
76
Zero-distribution
Partial integration gives
b
exp(2πi(f (u) − mu))g(u)
I1 (m) =
2πi
u=a
Z b
exp(2πif (u)) d
−
g(u) exp(−2πimu) du,
2πi
du
a
Z b
1
= O(g(a)) −
I2 (m) + m
g(u) exp(2πi(f (u) − mu)) du.
2πi
a
Thus,
X
f ′ (a)−η<m<f ′ (b)+η
m6=0
=
1
m
1
I1 (m) +
I2 (m)
2πi
Z
X
f ′ (a)−η<m<f ′ (b)+η
m6=0


+O 
b
g(u) exp(2πi(f (u) − hu)) du
a
X
f ′ (a)−η<m<f ′ (b)+η
m6=0

g(a) 
.
|m|
Now assume that m > f ′ (a) + η and f ′ (b) > 0. Then f ′ (u) > 0 for
a ≤ u ≤ b. Using Lemma 2.7 with F (u) = 2π(f (u) − mu) and G = gf ′, we
find
g(a)f ′ (a) .
I1 (m) ≪ ′
f (a) − m Hence,
I1 (m) m ≪ g(a)
′
m>f (a)+η
X
m6=0
X
0<m≤2|f ′ (a)|
1
+ g(a)
m
X
m>|f ′ (a)|
|f ′ (a)|
.
m2
The contribution arising from m < f ′ (b) − η can be treated similarly. This
gives
X
I1 (m) ′
′
m ≪ g(a) log(|f (a)| + |f (b)| + 2).
m6∈[f ′ (b)−η,f ′ (a)+η]
m6=0
Next assume m > f ′ (a) + η and m 6= 0. Then, by the mean-value theorem,
Z b
Re I2 (m) = −
|g ′(u)| cos 2π(f (u) − mu) du
a
′
= g (a)
Z
a
ξ
cos 2π(f (u) − mu) du
Section 2.2
The approximate functional equation
77
with some ξ ∈ (a, b). Partial integration yields
ξ
Z ξ
exp(2πi(f (u) − mu)
+
cos 2π(f (u) − mu) du = −Re
2πim
a
u=a
Z
1 ξ ′
f (u) exp(2πi(f (u) − mu)) du
+Re
m a
|f ′ (a)|
1
1+ ′
.
≪
|m|
|f (a) − m|
Therefore,
X
m>f ′ (a)+η
Re I2 (m) ≪ g ′ (a).
m
With slight modifications this method also applies to the cases Im I2 (m)
and m ≤ f ′ (b) − η. Further, if 0 6∈ [f ′ (b) − η, f ′ (a) + η], then Lemma 2.7 gives
Z b
g(u) exp(2πif (u)) du ≪ g(a).
a
In view of (2.6) the theorem follows from the above estimates under the
condition f ′ (b) > 0. If this condition is not fulfilled, then one can argue with
f (u) − ku, where k := 1 − [f ′ (b)], in place of f (u). •
2.2.3. Proof of the approximate functional equation. Now we apply van der Corput’s summation formula to the zeta-function. Let σ > 0. By
Theorem 1.3 we have
Z ∞
X exp(−it log n) N 1−s
X 1
[u] − u
+
+
+s
du.
ζ(s) =
s
σ
n
n
s−1
us+1
N
n≤x
x<n≤N
t
. Assume
Setting g(u) = u−σ and f (u) = − 2πt log u, we get f ′ (u) = − 2πu
1
7
′
that |t| ≤ 4x, then |f (u)| ≤ 8 . With the choice ǫ = 10 the interval
(f ′ (b) − η, f ′ (a) + η) contains only the integer m = 0. Thus, van der Corput’s
summation formula, Theorem 2.6, yields
Z N
X exp(−it log n)
=
u−s du + O(x−σ )
σ
n
x
x<n≤N
=
In addition with
s
Z
∞
N
N 1−s − x1−s
+ O(x−σ ).
1−s
[u] − u
du ≪ |s|N −σ
us+1
we deduce Theorem 2.3. •
Theorem 2.3 is is a first version of a family of formulae each of them called
approximate functional equation; the name reflects the appearance of the
quantities s and 1 − s as in the functional equation. There are stronger
78
Chapter 2
Zero-distribution
approximate functional equations known and their derivation relies heavily
on the functional equation; for instance:
Theorem 2.8. Let 0 ≤ σ ≤ 1 and x, y, t > C > 0, where C is a constant C
and 2πxy = t. Then
X 1
X 1
1
−σ
−σ σ−1
2
+ ∆(s)
+O x +t
y
(2.6)
ζ(s) =
ns
n1−s
n≤y
n≤x
uniformly in σ, where
πs
.
2
Here we have approximation by two shorter sums and with a much smaller
error term (if x and y are well balanced). This approximate functional equation was found by Hardy & Littlewood [78] in 1923 but was also known by
Riemann himself (see Siegel’s paper [182] on Riemann’s unpublished papers
on ζ(s)). The proof relies on complex variable methods, starting from the
identity
Z ∞ s−1
X 1
1
x exp(−mx)
+
dx
ζ(s) =
s
n
Γ(s)
exp(x)
−
1
0
n≤m
(2.7)
∆(s) := 2s π s−1 Γ(1 − s) sin
and contour integration; more details can be found in Ivić [98]. As a matter
of fact, this approach is very much tied to the functional equation for ζ(s).
An important extension of the classic techniques was given by Chandrasekharan & Narasimhan [30] for general Dirichlet series with functional
equations (e.g., Dedekind zeta-functions).
Exercise 58. Check that the function ∆(s) defined by (2.7) satisfies
ζ(s) = ∆(s)ζ(1 − s).
Exercise 59. Deduce from Theorem 2.3 that, for any fixed σ ∈ [ 12 , 1),
(2.8)
ζ(σ + it) ≪ t1−σ
as
t → ∞.
Can you improve this estimate by use of (2.6)?
Exercise 60. * Prove Theorem 2.8.
Hint: for this aim consult Ivić [98].
2.3. Power moments
Power moments are important tools in the theory of Dirichlet series; in
particular, they give information on the number of zeros as we shall see
below. We follow [200], §VII and §IX.
Section 2.3
Power moments
79
2.3.1. The quadratic mean. Our aim is the second moment. By use
of the approximate functional equation, we shall derive an asymptotic meansquare formula for ζ(s) with error term valid in the half-plane σ > 21 .
Theorem 2.9. For σ > 21 ,
Z T
|ζ(σ + it)|2 dt = ζ(2σ)T + O(T 2−2σ log T ).
1
Proof. By the approximate functional equation,
X 1
ζ(σ + it) =
+ O(t−σ ).
σ+it
n
n<t
Using this and ζ(σ − it) = ζ(σ + it), we get
2
Z T X
Z T X
1 1
dt
dt =
σ+it
σ+it
mσ−it
1 n<t n
1 m,n<t n
Z T it
X
m
1
dt
=
σ
(mn) τ
n
m,n<T
with τ := max{m, n}. The diagonal terms m = n give the contribution
!
X 1
X 1
XT −n
=
T
ζ(2σ)
−
−
n2σ
n2σ
n2σ−1
n≥T
n<T
n<T
= ζ(2σ)T + O(T 2−2σ ).
The non-diagonal terms m 6= n contribute
m iτ
m iT
X
X
−
1
1
n
n
≪
n
n .
σ
σ
(mn)
i log m
(mn) log m
m,n<T
0<m<n<T
m6=n
If 1 ≤ m <
n
,
2
n
then log m
> log 2 > 0, and hence
XX
n<T m< n
2
1
n
(mn)σ log m
≪
X 1
nσ
n<T
!2
≪ T 2−2σ .
If n2 ≤ m ≤ n, we write n = m + r with 1 ≤ r ≤ n2 . By the Taylor series
expansion of the logarithm,
r r
n
> .
= − log 1 −
log
m
n
n
This gives
XX
X
X1
1
1−2σ
≪ T 2−2σ log T.
≪
n
n
σ
(mn) log m
r
n
n
n<T
n<T
r≤ 2
r≤ 2
Collecting together, the assertion of the theorem follows. •
80
Chapter 2
Zero-distribution
The formula of Theorem 2.9 cannot hold for σ = 21 since then the main
term becomes singular: ζ(2σ) is unbounded as σ → 21 +. Indeed, on the
critical line the quadratic mean is of rather different form.
Theorem 2.10. As T → ∞,
2
Z T 1
1
ζ
2
+ it dt = T log T + O T (log T ) .
2
0
This result is due to Hardy & Littlewood [77]. For the proof we refer once
more to Ivić [98].
2.3.2. Higher moments. It is a long standing conjecture that for fixed
k ≥ 0, there exists a constant C(k) such that
Z
1 T
2
|ζ( 12 + it)|2k dt ∼ C(k)(log T )k ,
(2.9)
T 0
as T → ∞. It is not known whether this conjecture is related to Riemann’s
hypothesis or not. The asymptotic formula (2.9) is known to be true only in
the trivial case k = 0, and the cases k = 1 and k = 2 by the classical results
of Hardy & Littlewood [77] (Theorem 2.10) and Ingham [96] who showed
that
4
Z T 1
dt ∼ 1 T (log T )4 .
+
it
(2.10)
ζ
2
2π 2
0
This was improved by several authors who gave further main terms and
appropriate error terms. There is a different and remarkable approach of
Motohashi [152] to the fourth moment using the spectral theory of the nonEuclidean Laplacian on the upper half-plane.
Very little is known for higher moments. For the twelfth moment HeathBrown [84] gave the estimate
12
Z T ζ 1 + it dt ≪ T 2 (log T )17 .
2
0
By the work of Balasubramanian & Ramachandra [12] a lower bound of the
expected size holds for an arbitrary positive integer k
Z
1 T
2
|ζ( 12 + it)|2k dt ≫ (log T )k .
T 0
However, satisfying upper bounds are, even under assumption of Riemann’s
hypothesis, not known. For a nice introduction to these questions we refer
to Ivić’s monograph [98] and the survey Matsumoto [144].
Recently, Conrey & Gonek [39] and Keating & Snaith [114] stated a conjecture for the constant C(k) appearing in (2.9); remarkably, their heuristics
Section 2.3
Power moments
81
differ one from another (see also the survey Conrey [36]). To state this conjecture we define
2
k2 X
∞ Y
1
Γ(m + k)
1
.
(2.11)
a(k) =
1− 2
m
p
m!Γ(k)
p
m=0
p
Note that one has to take an appropriate limit if k is an integer less than
or equal to zero. It is not difficult to verify that a(1) = 1 and a(2) = π62 ;
however, further values are not explicitly known. Furthermore we have to
introduce Barnes’ double Gamma-function
G(z + 1) = (2π)z/2 exp − 12 (z(z + 1) + γz 2 ) ×
∞
Y
z2
z n
×
1 + n exp −z + n ,
n=1
where γ is Euler’s constant (there will be no confusion with the imaginary
parts of the zeros of ζ(s)); note that G(1) = 1 and G(z + 1) = Γ(z)G(z).
The approach of Conrey & Gonek [39] is of combinatorial nature. They
investigated mean-value theorems for Dirichlet polynomials and proved
2
Z a(k)
1 T X dk (n) 2
(log x)k
dt ∼
1/2+it
2
T 0 n≤x n
Γ(1 + k )
for x = o(T ), where dk (n) is the generalized divisor function appearing as
coefficients in the Dirichlet series representation of ζ(s)k . Assuming that the
limit

2 −1
Z T
Z T X
dk (n) g(k) := lim
|ζ( 21 + it)k |2 dt 
dt
1/2+it T →∞
0
0 n≤x n
exists, they were led to conjecture
C(k) =
a(k)g(k)
Γ(1 + k 2 )
for the constant in (2.9). Here one has g(1) = 1 and g(2) = 2. Furthermore,
they conjectured g(3) = 42 and g(4) = 24024. On the contrary, Keating
& Snaith [114] used the random matrix analogue. In fact, they proved, for
fixed k > − 21 ,
Z 2π
1
G(k + 1)2 k2
EN
|ZN (θ; U)|2k dθ = EN |ZN (0; U)|2k ∼
N ;
2π 0
G(2k + 1)
this corresponds to a continuous 2k-th moment of the characteristic polynomial ZN (θ; U) associated with an arbitrary matrix U from the unitary group
U(N) of all N × N matrices U with complex entries satisfying the condition
t
UU = idN ,
Chapter 2
82
Zero-distribution
t
where U denotes the transpose of the complex conjugate of U and idN is
the N × N identity matrix. The factor on the right-hand side of Keating &
Snaith’s formula was found to coincide with some data from the Conrey &
Gonek-approach, namely
k−1
Y
g(k)
j!
G(k + 1)2
=
=
,
Γ(1 + k 2 ) j=0 (j + k)!
G(2k + 1)
– what a surprise! The standard Random Matrix Theory-model cannot detect
the arithmetic factor (2.11): prime numbers do not occur in this model.
Consequently, the arithmetic information a(k), appearing in the heuristics of
Conrey & Ghosh, has to be inserted in an ad hoc way. Recently, Conrey,
Keating et al. modified the standard Random Matrix Theory-model which
incorporates also the arithmetic information a(k) (see Gonek [61]); this leads
directly to
Conjecture 1. For fixed k > − 12 , as T → ∞,
Z
2k
1 T 1
G(k + 1)2
2
ζ 2 + it dt ∼ a(k)
(log T )k .
T 0
G(2k + 1)
Needless to say that this conjecture includes the only known cases, the trivial
one k = 0, and the classical cases k = 1 and k = 2 due to Hardy & Littlewood
and Ingham, respectively.
2.3.3. The Lindelöf hypothesis. For many applications in number
theory it is useful to assume Riemann’s hypothesis but quite often it suffices to work with weaker conjectures. Lindelöf [134] conjectured that ζ(s)
is bounded if σ ≥ 21 + ǫ with any fixed positive ǫ. This would imply that
1
ζ
+ it ≪ tǫ
2
as t → ∞. The last statement is now known as Lindelöf’s hypothesis and
it is yet unproved. However, the strong boundedness conjecture is false (see
the related exercise below). The Lindelöf hypothesis follows from the truth
of the Riemann-hypothesis (as it follows from (2.22) below).
There are several further interesting reformulations of the Lindelöf hypothesis in case of the Riemann zeta-function. One, given in terms of moments
on the critical line, was found by Hardy & Littlewood [79]. They proved
that the Lindelöf hypothesis is true if and only if all power moments are
sufficiently small:
Section 2.3
Power moments
83
Theorem 2.11. The Lindelöf hypothesis is true if and only if, for any k ∈ N,
2k
Z 1
1 T ζ
+ it dt ≪ T ǫ .
T 1
2
A proof may be found in Titchmarsh [200]. This statement may serve as
a first example for the importance of power moment estimates. Further
examples will be given in the following sections.
Exercise 61. Show that
Z
T
0
ζ
1
+ it
2
1
dt ≪ T 2 log T.
Exercise 62. Prove a corresponding statement as Theorem 2.9 for Dirichlet Lfunctions.
Exercise 63. Prove Theorem 2.10. Try to obtain a better error term...
Hint: use an approximate functional equation.
Exercise 64. * Use Theorem 2.8 to prove
4
Z T ζ 1 + it dt ≪ T (log T )4
2
0
Hint: one may consult, e.g., Ivić [98].
Using the theory of diophantine approximations Harald Bohr (the brother of the
physicist Niels Bohr and medal winner in the olympic football team of Denmark
1908) & Landau [20] showed that ζ(s) takes arbitrarily large values in the halfplane of absolute convergence Re s > 1 and s not from the neighborhood of the
pole at s = 1.
Exercise 65. * i) Show that for σ > 1
ζ(σ) ≥ |ζ(s)| ≥
N
X
cos(t log n)
n=1
nσ
−
∞
X
n=N +1
1
.
nσ
ii) Prove the following statement about diophantine approximation (Dirichlet’s approximation theorem): Given arbitrary real numbers α1 , . . . , αN , a positive integer
q, and a positive number T , there exist real number τ ∈ [T, q N T ] and integers
x1 , . . . , xN for which
1
for n ≤ N.
|τ αn − xn | ≤
q
log n
2π
to find a real number τ ∈ [T, q N T ] such that
2π
for n = 1, . . . , N.
cos(τ log n) ≥ cos
q
iii) Apply ii) with αn =
iv) Prove the existence of an infinite sequence of s = σ + it with σ → 1+ and
t → ∞ for which
|ζ(s)| ≥ (1 − ǫ)ζ(σ),
84
Chapter 2
Zero-distribution
where ǫ is an y positive constant, and deduce that for arbitrary T > 0
lim sup |ζ(σ + it)| = ∞.
σ>1,t>T
Exercise 66. * Prove Theorem 2.11.
Hint: one may consult, e.g., Ivić [98].
2.4. Hardy’s theorem: zeros on the critical line
In 1914, Hardy [75] showed that there are indeed infinitely many zeros
of the Riemann zeta-function on the critical line. This was generalized by
Lekkerkerker [129] to a general class of Dirichlet series satisfying a Riemanntype functional equation.
2.4.1. Hardy’s Z-function. The behaviour of ζ(s) on the critical line
is reflected by Hardy’s Z-function Z(t) as a function of a real variable, defined
by
1
+ it ,
Z(t) = exp(iϑ(t))ζ
2
where
exp(iϑ(t)) := π
−it/2
Γ( 14 + it2 )
.
|Γ( 14 + it2 )|
It follows from the functional equation for ζ(s) that Z(t) is an infinitely often
differentiable function which is real for real t. Moreover,
ζ 1 + it = |Z(t)|.
2
Consequently, the zeros of Z(t) correspond to the zeros of the Riemann zetafunction on the critical line (counting multiplicities).
The function Z(t) has a negative local maximum at t = 2.4757 . . ., and
this is the only known negative local maximum in the range t ≥ 0; a positive
local minimum is not known. The occurrence of a negative local maximum,
besides the one at t = 2.4757 . . ., or a positive local minimum of Z(t), would
disprove Riemann’s hypothesis. Indeed, one can show that if the Riemann
hypothesis is true, the graph of the logarithmic derivative Z ′ /Z(t) is monotonically decreasing between the zeros of Z(t) for t ≥ 1000. A proof of this
claim can be found in Edwards [49].
Hardy’s Z-function allows to localize zeros on the critical line by applying
methods from real analysis. The Riemann-Siegel formula (discovered by Riemann, rediscovered by Siegel while studying Riemann’s unpublished papers)
Section 2.4
Hardy’s theorem
85
3
2
1
10
20
30
40
50
60
-1
-2
-3
Figure 1. Graphs of the modulus of the zeta-function (red) on the
critical line Re s =
1
2
and of Hardy’s Z-function (blue).
provides a very good approximation of the zeta-function on the critical line;
a first an rather primitive form is
X cos(ϑ(t) − t log n)
−1/4
(2.12)
Z(t) = 2
+
O
t
,
n1/2
√
n≤
t/(2π)
valid for t ≥ 1. We observe the similarity to approximate functional equations. The Riemann-Siegel formula is the basis of all high precision computations of the zeta-function on the critical line.1
Lehmer [128] detected that the zeta-function occasionally has two very
close zeros on the critical line; for instance the zeros at t = 7005.0629 . . . and
t = 7005.1006 . . .. So the graph of Z(t) sometimes barely crosses the t-axis
(see Figure 4).
In view of our observation relating the graph of Z ′ /Z(t) with Riemann’s
hypothesis from the previous section, Z(t) has exactly one critical point between successive zeros for sufficiently large t. Hence, Lehmer’s observation,
in the literature called Lehmer’s phenomenon, is a near-counterexample to
the Riemann hypothesis.
2.4.2. Hardy’s theorem. Now we are going to prove that there are
infinitely many nontrivial zeros of the zeta-function on the critical line. However, we shall sketch the proof of a quantitative version:
1A
very nice animated plot of Z(t) can be found on Pugh’s webpage
http://www.math.ubc.ca/ pugh/RiemannZeta/RiemannZetaLong.html.
Chapter 2
86
Zero-distribution
7005.02 7005.04 7005.06 7005.08
7005.1
7005.12 7005.14
-0.01
-0.02
-0.03
-0.04
-0.05
-0.06
Figure 2. Lehmer’s phenomenon.
1
Theorem 2.12. For sufficiently large T and H ≥ T 4 +ǫ , the interval (T, T +
H) contains at least one ordinate of a nontrivial zero ρ = β + iγ of ζ(s) (of
odd order).
Sketch of the proof. We shall compare the to integrals
Z T +H
Z T +H
I1 :=
|Z(t)| dt
and
I2 := Z(t) dt .
T
T
The main idea is rather simple. If I1 > I2 , there is a sign change for Z(t) in
the interval (T, T + H) and we are done: the value of t for which Z(t) crosses
the t-axis is the ordinate of a nontrivial zero of odd order.
First of all, we bound I1 from below. Clearly,
Z T +H 1
+ it dt .
(2.13)
I1 ≥ ζ
2
T
Using the approximate functional equation in the form of Theorem 2.8, we
find
1
X
X 1
1
− 12 −it
+it
2
n
n
ζ
+ it = 1 +
+ exp(iθ1 (t))
+ O t− 4 ,
2
√t
√t
2≤n≤
2π
where
n≤
2π
π
t
+ .
2πe 4
The constant term 1 gives the contribution H to the bound of the right-hand
side of (2.13) while all other terms strongly oscillate for sufficiently large H.
1
It follows that I1 ≫ H for H ≥ T 4 +ǫ . For I2 we apply the Riemann-Siegel
formula (2.12) in order to find an upper bound of order o(H). (For the details
we refer to Karatsuba [111].)
θ1 (t) = −t log
Section 2.5
Density theorems
87
In the meantime several important quantitative improvements of Hardy’s
theorem 2.12 were made. Selberg [179] was the first to prove that a positive
proportion of all zeros lies exactly on σ = 12 . Let N0 (T ) denote the number
of zeros ρ of ζ(s) on the critical line with imaginary part 0 < γ ≤ T . The
idea to use mollifiers to dampen the oscillations of |ζ( 12 + it)| led Selberg to
lim inf
T →∞
1
N0 (T + H) − N0 (T )
> 0,
N(T + H) − N(T )
27
as long as H ≥ T 2 +ǫ . Karatsuba [110] improved this result to H ≥ T 82 +ǫ
by some technical refinements. The proportion is very small, about 10−6 as
2
Min calculated; a later refinement by Zhuravlev gives after all 21
if H = T
(cf. Karatsuba [111], p.36). However, the localized zeros are not necessarily simple. By an ingenious new method, working with mollifiers of finite
length, Levinson [132] localized more than one third of the nontrivial zeros
of the zeta-function on the critical line, and as Heath-Brown [85] and Selberg
(unpublished) discovered, they are all simple. By optimizing the technique
Levinson himself and others improved the proportion 31 sligthly, but more
recognizable is Conrey’s idea in introducing Kloosterman sums. So Conrey
[35] was able to choose a longer mollifier to show that more than two fifths
of the zeros are simple and on the critical line; Bauer [14, 15] improved this
proportion slightly. The use of longer mollifiers leads to larger proportions.
Farmer [52] observed that if it is possible to take mollifiers of infinite length,
then almost all zeros lie on the critical line and are simple. In [189] Steuding
found a new approach (combining ideas and methods of Atkinson, Jutila and
Motohashi) to treat short intervals [T, T + H], i.e., H = o(T ); it was proved
that for H ≥ T 0.552 a positive proportion of the zeros of the zeta-function
with imaginary parts in [T, T + H] lie on the critical line and are simple.
Exercise 67. Prove all statements concerning Z(t) from Section 2.4.1, except the
Riemann-Siegel formula.
Exercise 68. * i) Verify all steps in the proof of Theorem 2.12.
1
ii) Try to prove that if H ≥ T 2 +ǫ , then any interval (T, T + H) with sufficiently
large T contains more than ≫ H many ordinates (of odd order).
2.5. Density theorems
Now we are going to study the complementary question: can we prove that
there are not too many zeros to the right of the critical line? In our studies
we shall frequently use the Riemann-von Mangoldt formula (1.12).
Chapter 2
88
Zero-distribution
2.5.1. Zeros off the critical line. Now we shall prove that most of the
nontrivial zeros of ζ(s) cannot lie too far from the critical line σ = 21 . This
observation is from Bohr & Landau [21], resp. Littlewood [137].
First of all, we need Littlewood’s lemma which relates the zeros of an
analytic function f (s) with a contour integral over log f (s).
Lemma 2.13 (Littlewood). Let A < B and let f (s) be analytic on R :=
{s ∈ C : A ≤ σ ≤ B, |t| ≤ T }. Suppose that f (s) does not vanish on the
right edge σ = B of R. Let R′ be R minus the union of the horizontal cuts
from the zeros of f in R to the left edge of R, and choose a single-valued
branch of log f (s) in the interior of R′ . Denote by ν(σ, T ) the number of
zeros ρ = β + iγ of f (s) inside the rectangle with β > σ including zeros with
γ = T but not those with γ = −T . Then
Z
Z B
log f (s) ds = −2πi
ν(σ, T ) dσ.
∂R
A
We
give a sketch of the simple proof.
Cauchy’s theorem implies
R
R∂R′ log f (s) ds = 0, and so the left-hand side of the formula of the lemma,
, is minus the sum of the integrals around the paths hugging the cuts.
∂R
Since the function log f (s) jumps by 2πi across each cut (assuming for simplicity that the zeros of f in
R R are simple and have different height; the
general case is no harder), ∂R is −2πi times the total length of the cuts,
which is the right-hand side of the formula in the lemma. For more details
we refer to Titchmarsh [200], §9.9, or Littlewood’s original paper [137].
Note that Littlewood’s lemma can be used, in addition with Stirling’s formula and some facts about entire functions, to prove the Riemann-von Mangoldt formula (1.12) (see Chapter 3).
Let N(σ, T ) denote the number of zeros ρ = β + iγ of ζ(s) with β >
σ, 0 < γ ≤ T (counting multiplicities). We apply Littlewood’s lemma to the
function f (s) = (s − 1)ζ(s) and the rectangle with corners 2 ± iT, σ0 ± iT
where σ0 ∈ ( 12 , 1). Note that f (s) is entire and its zeros correspond one-to-one
to the zeros of ζ(s).
Z
Z 1
2N(σ, T ) dσ =
log f (s) ds
2π
σ0
∂R
where we have ν(σ, T ) = 2N(σ, T ) since the zeros are symmetrically distributed with respect to the real axis. Now we want to remove the factor
s − 1. Applying Littlewood’s lemma once again, we get
Z
i
log(s − 1) ds = 2π(1 − σ0 ),
∂R
and so the contribution of the factor s − 1 in the last but one formula is
bounded. Taking into account ζ(s) = ζ(s) and that the integral over the
Section 2.5
Density theorems
89
zero-counting function is real-valued, we find
Z 1
Z T
Z T
2π
N(σ, T ) dσ =
log |ζ(σ0 + it)| dt −
log |ζ(2 + it)| dt
σ0
0
0
Z σ0
Z σ0
(2.14)
arg ζ(σ)) dσ
+
arg ζ(σ + iT ) dσ −
2
2
+O(1);
here we define log ζ(s) to be the principal branch on the positive real axis.
The main contribution in (2.14) comes from the first integral on the righthand side. The last integral does not depend on T and so it is bounded.
Moreover, we obtain
(
)
Z T
X 1 Z T
log |ζ(2 + it)| dt = Re
exp(−itk log p) dt
2k
kp
0
0
p,k
∞
X
1
≪ 1.
≪
n2
n=2
Now we estimate arg ζ(σ + iT ). We may assume that T is not the ordinate
of zero. Since arg ζ(2) = 0 and
Im ζ(s)
,
arg ζ(s) = arctan
Re ζ(s)
where
Re ζ(2 + it) =
∞
X
cos(it log n)
n=1
n2
we have by the argument principle
Z ∞
∞
X
1
du
≥1−
>1−
= 0,
2
n
u2
1
n=2
π
.
2
Now assume that Re ζ(σ + iT ) vanishes q times as 21 ≤ σ ≤ 2. Divide the
interval [ 21 + iT, 2 + iT ] into q + 1 parts, throughout each of which Re ζ(s) is
of constant sign. Hence, again by the argument principle, in each part the
variation of arg ζ(s) does not exceed π. This gives
3
1
| arg ζ(s)| ≤ q +
π
for σ ≥ .
2
2
| arg ζ(2 + iT )| ≤
Further, q is the number of zeros of the function
1
g(z) = (ζ(z + iT ) + ζ(z − iT )) = Re ζ(z + iT )
2
1
for Im z = 0 and 2 ≤ Re z ≤ 2. Thus, q ≤ n( 32 ), where n(r) is the number of
zeros of ζ(s) for |z − 2| ≤ r. Obviously,
Z 2
Z 2
Z 2
n(r)
dr
4
3
3
n(r)
log .
dr ≥
dr ≥ n
=n
3
3
r
r
2
r
2
3
0
2
2
Chapter 2
90
Zero-distribution
By Jensen’s formula 1.17 we obtain
Z 2π
Z 2
1
n(r)
dr =
log |ζ(2 + r exp(iθ))| dθ − log |ζ(2)|.
r
2π 0
0
In view of (2.8),
1
1
ζ(σ + it) ≪ t 2
as t → ∞, we find g(z) ≪ T 2 . Thus we obtain
Z 2
1
3
n(r)
≤
dr ≪ log T.
q≤n
4
2
log 3 0 r
This yields
arg ζ(σ + iT ) ≪ log T
1
uniformly for σ ≥ ,
2
and, consequently, the same bound holds by integration with respect to 12 ≤
σ ≤ 2. The restriction that T has not to be an imaginary part of a zero of
ζ(s) can be removed from considerations of continuity. Therefore, we may
replace (2.14) by
Z T
Z 1
1
log |ζ(σ0 + it)| dt + O(log T ).
(2.15)
N(σ, T ) dσ =
2π 0
σ0
Now we need a further analytic fact due to Jensen: Jensen’s inequality states
that for any continuous function f (u) on [a, b],
Z b
Z b
1
1
log f (u) du ≤ log
f (u) du
b−a a
b−a a
(for instance, this can be deduced from the arithmetic-geometric mean inequality, or see [199], §9.623). Hence, we obtain for any fixed σ0 > 21
Z T
Z T
T
1
2
log |ζ(σ + it)| dt ≤ log
|ζ(σ + it)| dt ≪ T
2
T 0
0
by applying Theorem 2.9. Thus,
Z 1
N(σ, T ) dσ ≪ T.
σ0
+ 12 (σ0 − 21 ), then 12 < σ1 < σ0 and we get
Z 1
Z σ0
2
1
N(σ, T ) ≪ T.
N(σ, T ) dσ ≤
N(σ0 , T ) ≤
σ0 − σ1 σ1
σ0 − 12 σ1
Let σ1 =
1
2
In view of (2.15) we have proved
Theorem 2.14. For any fixed σ > 12 ,
N(σ, T ) ≪ T.
Section 2.5
Density theorems
91
The theorem above is a first example of a so-called density theorem. By the
Riemann-von Mangoldt formula (1.12) we see that
1
(2.16)
N(σ, T ) = o(N(T ))
for σ > ,
2
so all but an infinitesimal proportion of the zeros of ζ(s) lie in the strip
1
− ǫ < σ < 21 + ǫ, however small ǫ may be!
2
2.5.2. The zero-detection method. We want to prove a stronger result due to Bohr & Landau [22].
Theorem 2.15. For any fixed σ in
1
2
< σ < 1,
N(σ, T ) ≪ T 4σ(1−σ) (log T )10 .
Proof. For 2 ≤ V ≤ T let N1 (σ, V ) count the zeros ρ = β + iγ of ζ(s) with
β ≥ σ and 12 V < γ ≤ V . Taking x = V in Theorem 2.3 we have
X 1
V 1−s
ζ(s) =
+
+ O V −σ
s
k
s−1
k≤V
1
V
2
for
mial
< t ≤ V and
1
2
≤ σ ≤ 1. Multiplying this with the Dirichlet polynoMX (s) :=
where X = V 2σ−1 , gives
X µ(m)
,
ms
m≤X
ζ(s)MX (s) = P (s) + R(s),
where
R(s) ≪ |MX (s)|V −σ
and
P (s) :=
with
(2.17)
a(n) :=
X a(n)
X µ(m) X 1
=
ms k≤V k s n≤XV ns
m≤X
X
µ(m) =
m|n
m≤X,n≤mV
1 if m = 1,
0 if 1 < n ≤ X.
Note that MX (s), as the truncated Dirichlet series of the reciprocal of ζ(s),
1
. We shall use P (s) as a zero-detector. Let s = ρ be a zero of
mollifies ζ(s)
the zeta-function with 12 V < γ ≤ V . Then,
X a(n) 1 ≤ + O(|MX (ρ)|V −β ),
ρ
n
X<n≤XV
X a(n) 2
1 ≪ + O(|MX (ρ)|2 V −2β ).
ρ
n X<n≤XV
Chapter 2
92
Zero-distribution
Then, summing up both sides of the latter inequality over all such N zeros
leads to


X a(n) 2
X

(2.18)
N1 (V ) ≪
+ |MX (ρ)|2 V −2σ  .
ρ
n
σ≤β≤1
X<n≤XV
1
2 V <γ≤V
Now we divide the interval [ 21 V, V ] into subintervals of length 1 of the form
[2m + n − 1, 2m + n], where n = 1, 2 and 14 V − 1 ≤ m ≤ 21 V . Then, we may
write
X
σ≤β≤1
1
2 V <γ≤V
≤
1
V
4
2
X
X
X
−1≤m≤ 12 V n=1 2m+n−1<γ≤2m+n
≤ 2 max
1≤n≤2
1
V
4
X
X
.
−1≤m≤ 12 V 2m+n−1<γ≤2m+n
In view of the Riemann-von Mangoldt formula (1.12) there are only ≪ log V
P
many zeros with 2m + n − 1 < γ ≤ 2m + n. Now let ′ρ denote the largest
of the related sums according to 2m + n − 1 < γ ≤ 2m + n. Then
X
σ≤β≤1
1 V <γ≤V
2
≪ log V
′
X
,
ρ
resp. in (2.18)
(2.19)
N1 (V ) ≪ log V
′
X
ρ

X

X<n≤XV

2 2
X
a(n) µ(m) −2σ 
.
+
V
ρ
n mρ m≤X
First of all we shall give a bound for
2
′ X
X
b(n) S(Y ) :=
,
ρ n
ρ
Y <n≤U
where U ≤ 2Y and V ≥ Y ≥ 1 and
(2.20)
b(n) ≪
X
1 =: d(n),
d|n
where d(n) is the divisor function. By partial summation, for fixed ρ = β +iγ,
X b(n) Z U
X b(n)
−β
=
.
C(u)
du
with
C(u)
:=
ρ
iγ
n
n
Y
Y <n≤U
Y <n≤u
Section 2.5
Density theorems
93
Applying the Cauchy-Schwarz inequality we obtain
Z U
X b(n) −β−1
|C(u)| du + Y −β |C(U)|,
≪ Y
ρ
n Y
Y <n≤U
2
Z U
X b(n) −2β−1
|C(u)|2 du + Y −2β |C(U)|2 .
≪ Y
ρ
n Y
Y <n≤U
This leads to
2
′ X
X
b(n)
−2σ
S(Y ) ≪ Y
,
iγ
n ρ
Y <n≤W
where W ≤ U. Since the distance of the imaginary parts of counted zeros
ρr = βr + iγr is ≥ 1, we can find
2
X
b(n)niγr+1 Y <n≤W
2
Z γr+1 X
≤
b(n)nit dt
γr
Y <n≤W
Z γr+1 X
X
it
it +2
b(n)n ·
b(m) log m · m dt.
γr
Y <n≤W
Y <m≤W
Summation over r and application of Cauchy-Schwarz yields
p
S(Y ) ≪ Y −2σ (I1 + I1 I2 ),
where
I1 :=
Z
V
1
V
2
2
X
b(n)nit dt ,
I2 :=
Y <n≤W
Z
V
1
V
2
2
X
b(n) log n · nit dt.
Y <n≤W
Taking (2.17) into account, |a(n)| satisfies condition (2.20) on b(n). By
elementary estimates one can show that
X
dk (n) ≪ x(log x)k ,
n≤x
where the implicit constant depends only on k; a proof can be found in [112]
(see also Exercise 33). This yields
X
I1 ≪ (V + Y ) log V
d2 (n) ≪ (V Y + Y 2 )(log V )5 ,
Y <n≤2Y
2
I2 ≪ (V Y + Y )(log V )7 .
Chapter 2
94
Zero-distribution
Now dividing the first sum on the right hand side of (2.19) into ≪ log V
sums, application of the latter estimates yields
2
′ X
X
a(n) log V
≪ (V X 1−2σ + (V X)2−2σ )(log V )9 .
nρ ρ
X<n≤V X
Similarly, we get for the second term
2
′ X
X
µ(m)
V −2σ (log T )2
≪ V −2σ (V + X 2−2σ )(log V )9 .
ρ
m ρ
m≤X
Substituting this in (2.19) with regard to X = V 2σ−1 , we obtain
N1 (V ) ≪ V 4σ(1−σ) (log V )9 .
Using this with V = T 1−n and summing up over all n ∈ N, proves the
theorem. •
2.5.3. The density hypothesis. There are stronger estimates known
than the one of Theorem 2.15. For instance, the strongest one which holds
throughout the right half of the critical strip is
N(σ, T ) ≪ T 2.4(1−σ) (log T )18.2
due to Huxley [94], resp. Gritsenko [70] who improved the former exponent
of the log-term. This estimate has remarkable consequences on the prime
number distribution; namely, it follows that
ψ(x + xθ ) − ψ(x) = xθ + o xθ
7
for any θ > 12
, as x → ∞. This implies that for sufficiently large x, there is
7
always a prime number p in any interval (x, x + x 12 +ǫ ), but it is too weak to
prove that there is always a prime in between consecutive squares.
The density hypothesis states that, for all σ > 12 ,
(2.21)
N(σ, T ) ≪ T 2(1−σ)+ǫ .
Of course, if the Riemann hypothesis is true, then N(σ, T ) is identically
zero for any σ > 12 . How is the density hypothesis related to the Lindelöf
hypothesis? Backlund [6] proved that the Lindelöf hypothesis is equivalent
to the much less drastic but yet unproved hypothesis that for every σ > 21
(2.22)
N(σ, T + 1) − N(σ, T ) = o(log T ).
Furthermore, the Lindelöf hypothesis implies the density hypothesis. Therefore, we have the following hierarchy:
Riemann hypothesis ⇒ Lindelöf hypothesis ⇒ density hypothesis.
Section 2.6
Universality
95
Exercise 69. Denote the zeros of ζ(s) by ρ = β + iγ. Show that, for fixed σ0 > 12 ,
X
(β − σ0 ) ≪ T.
0<γ≤T
β>σ0
Hint: compute the integral
R1
σ0
N (σ, T ) dσ.
Exercise 70. * i) Prove Backlund’s statement that (2.22) is equivalent to the
Lindelöf hypothesis.
ii) Show that the Lindelöf hypothesis implies the density hypothesis.
Hint: one may consult, for example, Patterson [164].
2.6. Universality and self-similarity.
We conclude with an application of the results presented in the last sections,
Voronin’s famous universality theorem [206] which roughly states that any(!)
non-vanishing analytic function can be approximated uniformly by certain
shifts of the Riemann zeta-function. This universal property is related to the
zero-distribution; we shall deduce an equivalent for the truth of the Riemann
hypothesis due to Bohr and Bagchi.
2.6.1. Voronin’s universality theorem. In 1975, Voronin [206]
proved the following
Theorem 2.16. Let f (s) be a non-vanishing continuous function defined on
a disk {s ∈ C : |s| ≤ r} with some r ∈ (0, 41 ), and analytic in the interior.
Then, for any ǫ > 0, there exists τ > 0 such that
3
max ζ s + + iτ − f (s) < ǫ;
|s|≤r
4
moreover,
1
lim inf meas
T →∞ T
3
τ ∈ [0, T ] : max ζ s + + iτ − f (s) < ǫ > 0.
|s|≤r
4
Thus, the set of τ for which shifts of the zeta-function approximate f (s) with
a given accuracy has positive lower density (with respect to the Lebesgue
measure). We say that ζ(s) is universal since appropriate shifts approximate
uniformly any element of a huge class of functions.
We give a very brief sketch of Voronin’s argument following the book of
Karatsuba & Voronin [112]. The Euler product for ζ(s) is the key to prove
the universality theorem in spite of the fact that it does not converge in the
region of universality. However, as Bohr observed, an appropriate truncated
Euler product approximates ζ(s) in a certain mean-value sense inside the
critical strip; this is related to the use of modified truncated Euler products
in Voronin’s proof (see (2.25) and (2.26) below).
Chapter 2
96
Zero-distribution
It is more convenient to work with series than with products. Therefore, we
consider the logarithms of the functions in question. Since g(s) has no zeros
in |s| ≤ r its logarithm exists and we may define an analytic function f (s)
on |s| ≤ r by g(s) = exp f (s). First we approximate f (s) by the logarithm
of a truncated Euler product. Let Ω denote the set of all sequences of real
numbers indexed by the prime numbers in ascending order. Further, define
for every finite subset M of the set of all primes, every ω = (ω2 , ω3 , . . .) ∈ Ω
and all complex s,
−1
Y
exp(−2πiωp )
.
1−
ζM (s, ω) =
ps
p∈M
Obviously, ζM (s, ω) is a non-vanishing analytic function of s in the half-plane
σ > 0. Consequently, its logarithm exists and is equal to
X
exp(−2πiωp )
;
log ζM (s, ω) = −
log 1 −
s
p
p∈M
in order to have a definite value we may choose the principal branch of the
logarithm. Since f (s) is uniformly continuous in the disc |s| ≤ r, there exists
some κ > 1 such that κ2 r < 41 and
s
ǫ
max f
− f (s) < .
2
|s|≤r
κ
2
The function f κs2 is bounded on the disc |s| ≤ κr =: R, and thus belongs
2
to the Hardy space HR
, i.e., the Hilbert space consisting of those functions
F (s) which are analytic for |s| < R with finite norm
ZZ
|F (s)| dσ dt
kF k := lim
r→R−
|s|≤r
and inner product
hF, Gi := Re
ZZ
F (s)G(s) dσ dt.
|s|≤R
Denote by pk the k-th prime number. We consider the series
∞
X
k=1
uk (s, ω),
where uk (s, ω) := log 1 −
exp(−2πiωpk )
s+ 43
pk
!−1
.
Here comes the first main idea. Riemann proved that any conditionally convergent series can be rearranged such that its sum converges to an arbitrary
preassigned real number. Pechersky [165] generalized Riemann’s theorem to
Hilbert spaces. It follows, with the special choice ω = ω0 = ( 41 , 42 , 34 , . . .), that
P
there exists a rearrangement of the series
uk (s) for which
∞
s
X
ukj (s, ω0 ) = f
κ2
j=1
Section 2.6
Universality
97
(the rather difficult and lengthy verification of the conditions of Pechersky’s theorem uses classic results of Paley & Wiener and Plancherel from
Fourier analysis, a theorem on the approximation by polynomials due to
A.A. Markov, and, most importantly, the prime number theorem 1.9).
The tail of the rearranged series can be made as small as we please, say of
modulus less than 2ǫ . Thus, it turns out that for any ǫ > 0 and any y > 0
there exists a finite set M of prime numbers, containing at least all primes
p ≤ y, such that
3
max log ζM s + , ω0 − f (s) < ǫ.
(2.23)
|s|≤r
4
The next and main step in Voronin’s proof is to switch from log ζM (s) to
the logarithm of the zeta-function. Of course, log ζ(s) has singularities at
the zeros of ζ(s), but since the set of these possibly singularities has measure
zero by density theorem 2.15, they are negligible.
We choose κ > 1 and ǫ1 ∈ (0, 1) such that κr < 41 and
s
(2.24)
max f
− f (s) < ǫ1 .
|s|≤r
κ
Putting Q = {p : p ≤ z} and E = {s = σ + it : −κr < σ ≤ 2, |t| ≤ 1}, one
can show, using the approximate functional equation for ζ(s), Theorem 2.3,
that for any ǫ2 > 0
(2.25)
2
Z 2T Z Z −1
3
3
dσ dt dτ ≪ ǫ4 T,
ζ
+
iτ,
0
ζ
s
+
+
iτ
−
1
s
+
2
Q
4
4
T
E
provided that z and T are sufficiently large, depending on ǫ2 ; here 0 :=
(0, 0, . . .). Now define
(
)
ZZ
AT = τ ∈ [T, 2T ] :
|ζQ−1(s + iτ, 0)ζ(s + iτ ) − 1|2 dσ dt < ǫ22 .
E+ 34
Then it follows from (2.25) that, for sufficiently large z and T ,
(2.26)
meas (AT ) > (1 − ǫ2 )T,
which is surprisingly large. It follows from Cauchy’s formula that, for sufficiently small ǫ2 ,
3
3
(2.27) max log ζ s + + iτ − log ζQ s + + iτ, 0 ≪ ǫ2 ,
|s|≤r
4
4
provided τ ∈ AT , where the implicit constant depends only on κ. By (2.23)
there exists a sequence of finite sets of prime numbers M1 ⊂ M2 ⊂ . . . such
that ∪∞
k=1 Mk contains all primes and
s 3
= 0.
lim max log ζMk s + , ω0 − f
k→∞ |s|≤κr 4
κ Chapter 2
98
Zero-distribution
(0)
(0)
Let ω0 = (ω2 , ω3 , . . .). By the continuity of log ζM s + 43 , ω0 , for any
ǫ1 > 0 there exists a positive δ such that, whenever the inequalities
kωp(0) − ωp k < δ
(2.28)
for p ∈ Mk
hold, where kzk denotes the minimal distance of z to an integer, then
3
3
max log ζMk s + , ω0 − log ζMk s + , ω < ǫ1 .
(2.29)
|s|≤κr
4
4
Let
BT =
log p
(0) − ωp < δ .
τ ∈ [T, 2T ] : τ
2π
Now we consider
Z ZZ
3
1
3
2
log ζQ s + + iτ, 0 − log ζMk s + + iτ, 0 dσ dt dτ,
T BT
4
4
|s|≤κr
resp.
ZZ
|s|≤κr
1
T
Z
Putting
BT
3
3
2
log ζQ s + + iτ, 0 − log ζMk s + + iτ, 0 dτ dσ dt.
4
4
log 2 log 3
,τ
,... ,
ω(τ ) = τ
2π
2π
(2.30)
we may rewrite the inner integral as
2
Z 3
3
log ζQ s + , ω(τ ) − log ζM s + , ω(τ ) dτ.
k
4
4
BT
Now we need Weyl’s refinement of Kronecker’s approximation theorem. Let
ω(τ ) be a continuous function with domain of definition [0, ∞) and range RN .
Then the curve ω(τ ) is said to be uniformly distributed mod 1 in RN if, for
Q
every parallelepiped
= [α1 , β1 ] × . . . × [αN , βN ] with 0 ≤ αj < βj ≤ 1 for
1 ≤ j ≤ N,
N
n
o Y
Y
1
meas τ ∈ (0, T ) : ω(τ ) ∈
mod 1 =
(βj − αj ).
T →∞ T
j=1
lim
In a sense, a curve is uniformly distributed mod 1 if the correct proportion
of values lies in a given subset of the unit cube. In questions about uniform
distribution mod 1 one is interested in the fractional part only. For a curve
ω(τ ) in RN , we define
{ω(τ )} = (ω1 (τ ) − [ω1 (τ )], . . . , ωN (τ ) − [ωN (τ )]),
where [x] denotes the integral part of x ∈ R.
Section 2.6
Universality
99
Lemma 2.17. i) Let a1 , . . . , aN be real numbers, linearly independent over Q,
and let γ be a subregion of the N-dimensional unit cube with Jordan content
Γ. Then
1
meas {τ ∈ (0, T ) : (τ a1 , . . . , τ aN ) ∈ γ mod 1} = Γ.
T →∞ T
lim
ii) Suppose that the curve ω(τ ) is uniformly distributed mod 1 in RN . Let D
be a closed and Jordan measurable subregion of the unit cube in RN and let
Ω be a family of complex-valued continuous functions defined on D. If Ω is
uniformly bounded and equicontinuous, then
1
lim
T →∞ T
Z
0
T
f ({ω(τ )})1D (τ ) dτ =
Z
f (x) dx
D
uniformly with respect to f ∈ Ω, where 1D (τ ) is equal to 1 if ω(τ ) ∈ D mod 1,
and zero otherwise.
Note that the notion of Jordan content is more restrictive than the notion of
Lebesgue measure. But, if the Jordan content exists, then it is also defined
in the sense of Lebesgue and equal to it. A proof of Weyl’s theorem can be
found in Karatsuba & Voronin [112].
The unique prime factorization of integers implies the linear independence
of the logarithms of the prime numbers over the field of rational numbers. By
Lemma 2.17, i), the curve ω(τ ), defined by (2.30), is uniformly distributed
mod 1. Application of Lemma 2.17, ii), to the curve ω(τ ) yields
1
lim
T →∞ T
Z
2
log ζQ s + 3 , ω(τ ) − log ζM s + 3 , ω(τ ) dτ
k
4
4
BT
2
Z 3
3
dµ,
,
ω
−
log
ζ
,
ω
s
+
log
ζ
s
+
=
M
Q
k
4
4
D
uniformly in s for |s| ≤ κr, where D is the subregion of the unit cube in
RN given by the inequalities (2.28) with N = ♯Mk , and dµ is the Lebesgue
measure. By the definition of ζM (s, ω) it follows that for Mk ⊂ Q
ζQ (s, ω) = ζMk (s, ω)ζQ\Mk (s, ω),
and thus
2
Z log ζQ s + 3 , ω − log ζM s + 3 , ω dµ
k
4
4
D
2
Z
3
≪ meas (D) ·
log ζQ\Mk s + , ω dµ.
4
[0,1]N
100
Chapter 2
Zero-distribution
2κr− 1
The latter integral is bounded above by yk 2 provided that Mk contains all
primes ≤ yk . It follows that
2
Z ZZ
1
log ζQ s + 3 + iτ, 0 − log ζM s + 3 + iτ, 0 dσ dt dτ
k
T BT
4
4
|s|≤κr
2κr− 21
≪ yk
meas (D).
Applying Lemma 2.17, ii) once more yields
1
meas (BT ) = meas (D),
T →∞ T
which implies, for sufficiently large yk ,
ZZ
n
3
meas τ ∈ BT :
log ζQ s + 4 + iτ, 0
|s|≤κr
2
o
3
κr− 1
− log ζMk s + + iτ, 0 dσ dt < yk 4
4
meas (D)
T,
>
2
and
n
3
meas τ ∈ BT : max log ζQ s + + iτ, 0
|s|≤κr
4
o
1
3
(κr− 41 )
− log ζMk s + + iτ, 0 < yk5
4
meas (D)
(2.31)
T.
>
2
If we now take 0 < ǫ2 < 21 meas (D), then (2.26) implies
lim
meas (AT ∩ BT ) > 0.
Thus, in view of (2.23) and (2.24) we may approximate f (s) by
log ζMk s + 43 , ω0 (independent on τ ), with (2.29) and (2.31) the latter function by log ζQ s + 43 + iτ, 0 , and finally with regard to (2.27) by
log ζ s + 43 + iτ on a set of τ with positive measure. Replacing T by 21 T ,
we thus find, for any ǫ > 0,
3
1
lim inf meas τ ∈ [0, T ] : max log ζ s + + iτ − f (s) < ǫ > 0.
T →∞ T
|s|≤r
4
Now taking the exponential we obtain Voronin’s theorem.
Theorem 2.16 was generalized and extended in several directions. Reich
[174] and Bagchi [7] replaced the disk by an arbitrary compact subset of the
right half of the critical strip with connected complement, and by giving a
lucid proof in the language of probability theory. The strongest version of
Voronin’s theorem has the form:
Section 2.6
Universality
101
Theorem 2.18. Suppose that K is a compact subset of the strip 21 < σ <
1 with connected complement, and let g(s) be a non-vanishing continuous
function on K which is analytic in the interior of K. Then, for any ǫ > 0,
1
lim inf meas τ ∈ [0, T ] : max |ζ(s + iτ ) − g(s)| < ǫ > 0.
s∈K
T →∞ T
This theorem can be found in the monograph Laurinčikas [127] which also
contains proofs of universality for Dirichlet L-functions; in Steuding [191]
universality for a large class of L-functions was proved.
A natural question arises: is the condition on the non-vanishing of g(s) in
the universality theorem necessary, i.e., is it possible to approximate uniformly
functions having zeros by shifts of ζ(s) (in the sense of Voronin’s universality
theorem)? The answer is negative. We give a heuristic argument which can
easily be made waterproof. It relies on the classic Rouché’s theorem:
Lemma 2.19. Let f (s) and g(s) be analytic for |s| ≤ r. If
|f (s) − g(s)| < |g(s)|
on |s| = r, then f (s) and g(s) have the same number of zeros in |s| < r.
This result follows from a simple application of the argument principle; for
details see Burckel [28], §VIII.3, or Titchmarsh [199], §3.42.
Assume that g(s) is an analytic function on |s| ≤ r, where 0 < r < 41 ,
which has a zero ξ with |ξ| < r but which is non-vanishing on the boundary.
An application of Rouché’s theorem shows that whenever the inequality
3
(2.32)
max ζ s + + iτ − g(s) < min |g(s)|
|s|=r
|s|≤r
4
holds, ζ s + 34 + iτ has to have a zero inside |s| < r. The zeros of an analytic
function lie either discretely distributed or the function vanishes identically
and thus the inequality (2.32) holds if the left hand side is sufficiently small.
If now for any ǫ > 0
3
1
lim inf meas τ ∈ [0, T ] : max ζ s + + iτ − g(s) < ǫ > 0,
T →∞ T
|s|≤r
4
then we expect ≫ T many complex zeros of ζ(s) in the strip 43 −r < σ < 34 +r
up to T (for a rigorous proof one has to consider the densities of values τ
satisfying (2.32); this can be done along the lines of the proof of Theorem
2.20 below). This contradicts the density theorem 2.15, which gives
3
− r, T = o(T ).
N
4
Thus, uniform approximation of a function g(s) having zeros by the zetafunction cannot be done!
102
Chapter 2
Zero-distribution
2.6.2. Almost periodicity. Bohr introduced the fruitful notion of almost periodicity into analysis. An analytic function f (s), defined on some
vertical strip a < σ < b, is called almost periodic if, for any positive ε, and
any α, β with a < α < β < b, there exists a length ℓ = ℓ(f, α, β, ε) > 0 such
that every interval (t1 , t2 ) of length ℓ contains an almost period of f relatively
to ε in the closed strip α ≤ σ ≤ β, i.e., there exists a number τ ∈ (t1 , t2 ) such
that
(2.33)
|f (σ + it + iτ ) − f (σ + it)| < ε
for α ≤ σ ≤ β, t ∈ R.
Bohr [19] proved that every Dirichlet series is almost-periodic in its halfplane of absolute convergence. Furthermore, he discovered an interesting
relation between the Riemann hypothesis and almost periodicity; indeed,
his aim in introducing the concept of almost periodicity might have been
Riemann’s hypothesis. Bohr showed that if χ is non-principal, then the
Riemann hypothesis for the Dirichlet L-function L(s, χ) is equivalent to the
almost periodicity of L(s, χ) in σ > 21 . The condition on the character might
appear to be a bit unnatural but is necessary for Bohr’s reasoning. His
argument relies in the main part on diophantine approximation applied to
Dirichlet series inside the critical strip. The Dirichlet series for L(s, χ) with
a non-principal character χ converges throughout the critical strip, but the
one for the zeta-function does not.
2.6.3. An equivalent for RH. More than half a century later Bagchi
[7] proved that the Riemann hypothesis is true if and only if the zeta-function
can approximate itself in the sense of Voronin’s universality theorem. In [8],
Bagchi generalized this result in various directions; in particular for Dirichlet L-functions to arbitrary characters. One implication of his proof in [8]
relies essentially on Voronin’s universality theorem (resp. its generalization
to Dirichlet L-functions), which, of course, was unknown to Bohr. Later,
Bagchi [9] gave another proof in the language of topological dynamics, independent of universality, and therefore this property, equivalent to Riemann’s
hypothesis, is called strong recurrence.
Theorem 2.20. Let θ ≥ 21 . Then ζ(s) is non-vanishing in the half-plane
σ > θ if and only if, for any ǫ > 0, any z with Re z > θ, and any 0 < r <
min{Re z − θ, 1 − Re z},
1
lim inf meas τ ∈ [0, T ] : max |ζ(s + iτ ) − ζ(s)| < ǫ > 0.
T →∞ T
|s−z|≤r
Proof. If Riemann’s hypothesis is true we can apply Voronin’s universality
theorem in the form of Theorem 2.18 with g(s) = ζ(s), which implies the
strong reccurence. The idea for the proof of the other implication is that
if there is at least one zero to the right of the line σ = θ, then the strong
Section 2.6
Universality
103
recurrence property implies the existence of many zeros, too many with regard
to the classic density theorem 2.15.
Suppose that there exists a zero ξ of ζ(s) with Re ξ > θ. Without loss of
generality we may assume that Im ξ > 0. We have to show that there exists
a disc with center z and radius r, satisfying the conditions of the theorem,
and a positive ǫ such that
1
(2.34)
lim inf meas τ ∈ [0, T ] : max |ζ(s + iτ ) − ζ(s)| < ǫ = 0.
T →∞ T
|s−z|≤r
Locally, the zeta-function has the expansion
(2.35)
ζ(s) = c(s − ξ)m + O |s − ξ|m+1
with some non-zero c ∈ C and m ∈ N. Now assume that for a neighbourhood
Kδ := {s ∈ C : |s − ξ| ≤ δ} of ξ the relation
(2.36)
max |ζ(s + iτ ) − ζ(s)| < ǫ ≤ min |ζ(s)|
s∈Kδ
|s|=δ
holds; the second inequality holds for sufficiently small ǫ. Then Rouché’s
theorem 2.19 implies the existence of a zero ρ of ζ(s) in
Kδ + iτ := {s ∈ C : |s − iτ − ξ| ≤ δ}.
We may say that the zero ρ of ζ(s) is generated by the zero ξ. With regard
to (2.35) and (2.36) the zeros ξ and ρ = β + iγ are intimately related; more
precisely,
ǫ > |ζ(ρ) − ζ(ρ − iτ )| = |ζ(ρ − iτ )| ≥ |c| · |ρ − iτ − ξ|m + O(δ m+1 ).
Hence,
|ρ − iτ − ξ| ≤
In particular,
and
ǫ
|c|
m1
1
+ O δ 1+ m .
m1
1
ǫ
< β < 1,
< Re ξ − 2
2
|c|
|γ − (τ + Im ξ)| < 2
ǫ
|c|
m1
,
for sufficiently small ǫ and δ = o(ǫm+1 ). Next we have to count the generated
zeros in terms of τ . Two different shifts τ1 and τ2 can lead to the same zero
ρ, but their distance is bounded by
m1
ǫ
.
|τ1 − τ2 | < 4
|c|
Chapter 2
104
Zero-distribution
If we now write
[
I(T ) :=
Ij (T ) := τ ∈ [0, T ] : max |ζ(s + iτ ) − ζ(s)| < ǫ ,
s∈Kδ
j
where the Ij (T ) are disjoint intervals, it follows that there are
#
" 1
1
1 |c| m
1 |c| m
meas (Ij (T )) + 1 >
meas (Ij (T ))
≥
4 ǫ
4 ǫ
many distinct zeros according to τ ∈ Ij (T ), generated by ξ. The number of
generated zeros is a lower bound for the number of all zeros. It follows that
(
m1 )
m1
ǫ
ǫ
, 0 < γ < T + Im ξ + 2
♯ ρ = β + iγ : β > Re ξ − 2
|c|
|c|
1
1 |c| m
meas (I(T )).
≥
4 ǫ
This and the density theorem 2.15 lead to
meas (I(T )) = o(T ),
which implies (2.34). The theorem is proved. •
The expected strong reccurence of ζ(s) may be regarded as a kind of selfsimilarity. Assuming the truth of Riemann’s hypothesis this has a nice interpretation. Consider the amplitude of light which is a physical bound for
the size of objects which human beings can see, or the Planck constant 10−33
which is the smallest size of objects in quantum mechanics. Thus, if we assume that ǫ is less than one of these quantities, then we cannot physically
distinguish between ζ(s) and ζ(s + iτ ) for s from a compact subset K of the
right half of the critical strip, whenever
max |ζ(s + iτ ) − ζ(s)| < ǫ.
s∈K
This shows that we cannot decide where we actually are in the analytic
landscape of ζ(s) without moving to the boundary. The zeta-function is an
amazing maze!
Exercise 71. * Study the proof of Voronin’s universality theorem in detail. Extend
the argument to Dirichlet L-functions.
CHAPTER 3
Modular forms and Hecke theory
This chapter is devoted to functional equations. We will prove the functional equation for the Riemann zeta-function and sketch the proof of the one
for Dirichlet L-functions. Furthermore, we will discuss in detail an important
link between Dirichlet series satisfying a Riemann-type functional equation
and modular forms, discovered by Hecke in the 1930s.
3.1. The functional equation for zeta and more
Now we shall prove the functional equation for Riemann’s zeta-function;
this will complete our studies on the analytic continuation from the first
chapter.
Theorem 3.1. For any s ∈ C,
s
1−s
− 1−s
− 2s
2
ζ(s) = π
ζ(1 − s).
Γ
π Γ
2
2
Riemann [175] himself gave two proofs of the functional equation. In the
meantime, quite many different proofs were found (see for example [200]).
Here we follow Riemann’s original approach which relies on the functional
equation of the theta-function. In order to give a rigorous proof we therefore
prove first Poisson’s summation formula and apply this to the theta-function
in order to obtain its functional equation. This is by far not the fastest way
to prove Theorem 3.1; however, this method applies to Dirichlet L-functions
as well and we shall sketch the proof of their functional equation too. But
more than that: this approach will also play a substantial role in the sequel
of this chapter.
3.1.1. The Poisson summation formula. Suppose f : R → C is an
integrable function satisfying f (z) ≪ |z|−2 as |z| → ∞ (actually, this is a
strong restriction but it allows to do the next step). Then we may define its
Fourier transform by
Z +∞
fˆ(y) =
f (z) exp(−2πiyz) dz.
−∞
The Poisson summation formula is a useful tool in Fourier theory with many
applications in real and complex analysis.
105
Chapter 3
106
Hecke theory
Theorem 3.2. Let f : R → R be a twice continuously differentiable function
with f (z) ≪ |z|−2 as |z| → ∞. Further, assume that the integral
Z +∞
|f ′′ (z)| dz
−∞
exists. Then, for any α ∈ R,
X
X
f (n + α) =
fˆ(m) exp(2πiαm).
n∈Z
m∈Z
Proof. It suffices to prove the formula in question only for α = 0. In fact,
writing g(z) = f (z + α) for fixed α ∈ R, we have ĝ(y) = fˆ(y) exp(2πiαy).
Therefore, we may assume α = 0.
First of all, for r > 0, define
∞
X
P (y, r) =
r |m| exp(2πimy).
m=−∞
This series is the sum of the term for m = 0 plus two infinite geometric
series, one for m < 0 and one for m > 0, both being absolutely convergent
for r ∈ [0, 1). Hence, we can compute the value of the infinite series P (y, r)
by
r exp(−2πiy)
r exp(2πiy)
+
1 − r exp(2πiy) 1 − r exp(−2πiy)
1 − r2
=
.
1 − 2r cos(2πy) + r 2
P (y, r) = 1 +
This implies P (y, r) ≥ 0 for any y (since the denominator is equal to (r −
cos 2πy)2 + (sin 2πy)2 ). Using
Z 1
1
if m = 0,
exp(2πimy) dy =
0
otherwise,
0
we find
Z
1
P (y, r) dy = 1
0
for all r ∈ [0, 1). Further note that P (y, r) is 1-periodic with respect to y.
Hence,
P (y, r) ≤
for 0 < δ ≤ |y| ≤ 21 .
1 − r2
(sin 2πδ)2
Section 3.1
The functional equation
Since f (z) ≪ z −2 , we have
+∞
X
r
|m|
Z
fˆ(m) =
m=−∞
+∞
P (y, r)f (y) dy
−∞
+∞ Z
X
=
107
P (y, r)f (y) dy;
[m− 21 ,m+ 12 ]
m=−∞
interchanging summation and integration is justified with respect to the absolute convergence. We want to show that the right-hand side converges to
P
m f (m) as r → 1−. For this purpose we note that
Z 1
Z
P (y, r) dy
P (y, r)f (y) dy ≤
max 1 |f (y)|
1
[m− 21 ,m+ 21 ]
m− 2 ≤y≤m+ 2
≤
max
m− 12 ≤y≤m+ 12
0
|f (y)| ≪ m−2 ,
as |m| → ∞. Hence, given ǫ > 0, there exists M > 0 such that
X
X Z
P (y, r)f (y) dy < ǫ
and
|f (m)| < ǫ.
|m|>M
[m− 21 ,m+ 21 ]
|m|>M
Now assume |m| ≤ M. Of course,
Z
Z
P (y, r)f (y) dy − f (m) =
[m− 12 ,m+ 21 ]
[m− 12 ,m+ 21 ]
P (y, r)(f (y) − f (m)) dy.
ǫ
for all m with |m| ≤ M and
Take some δ > 0 for which |f (y) − f (z)| < 3M
all y, z with |y − z| ≤ δ. Then
X Z
P (y, r)f (y) dy − f (m)
[m− 1 ,m+ 1 ]
2
2
|m|≤M
X
(3.1)
≤
(J1 (m) + J2 (m)),
|m|≤M
where
J1 (m) :=
Z
m+δ
m−δ
J1 (m) :=
Z
W (m)
P (y, r)|f (y) − f (m)| dy,
P (y, r)|f (y) − f (m)| dy
with W (m) := {y ∈ R : δ < |y − m| ≤ 12 }. By construction,
Z m+δ
ǫ
ǫ
P (y, r) dy ≤
.
J1 (m) ≤
3M m−δ
3M
Moreover,
1 − r2
J2 (m) ≤
(sin 2πδ)2
Z
W (m)
|f (y) − f (m)| dy ≪
1 − r2
,
m2
Chapter 3
108
Hecke theory
where the implicit constant depends only on δ, ǫ and f . Thus, the right-hand
side of (3.1) can be made less than 2ǫ for some r sufficiently close to 1. Hence,
letting ǫ → 0, we obtain
(3.2)
lim
r→1−
+∞
X
r
|m|
fˆ(y) =
m=−∞
+∞
X
f (m).
m=−∞
Partial integration shows fˆ(m) ≪ m−2 . Consequently, the series on the lefthand side of (3.2) converges absolutely and uniformly for r ∈ [0, 1) and we
may interchange summation and take the limit. This proves the theorem. •
3.1.2. The theta-function. The (most simple) theta-function is given
by the infinite series
X
θ(x) =
exp(−πxn2 ).
n∈Z
We apply Poisson’s summation formula, Theorem 3.2, to the function f (z) :=
exp(−πz 2 /x) with x > 0. We compute the Fourier transform by quadratic
substitution:
Z +∞
fˆ(y) =
exp(−π(z 2 /x + 2iyz)) dz
−∞
2
= x exp(−πxy )
(3.3)
Next we consider the integral
Z
I(λ) :=
Z
+∞
exp(−πx(w + iy)2 ) dw.
−∞
+∞
exp(−πx(w + λ)2 ) dw,
−∞
where λ is any complex number. Consider the integral
Z
exp(−xω 2 ) dω,
R
where R is the rectangular contour with vertices ±r, ±r + iIm λ, where r is a
positive real number. By Cauchy’s theorem, the integral is equal to zero. On
the line Re ω = r, the integrand tends uniformly to zero as r → ∞. Hence,
I(λ) = I(0), and thus the integral I(λ) does not depend on λ. This gives in
(3.3)
Z +∞
√
2
ˆ
f (y) = x exp(−πxy )
exp(−πxw 2 ) dw = C x exp(−πxy 2 ),
−∞
where
C :=
Z
+∞
exp(−πz 2 ) dz.
−∞
Applying Poisson’s summation formula leads to
X
√ X
exp(−π(n + α)2 /x) = C x
exp(−πxn2 + 2πinα);
n∈Z
n∈Z
Section 3.1
The functional equation
109
here we have introduced the parameter α by the trick from the proof of
Theorem 3.2. Choosing α = 0 and x = 1, both sums are equal; thus, C = 1
and we have just proved the functional equation for the theta-function:
Theorem 3.3. For any x > 0,
1
θ(x) = √ θ
x
1
.
x
3.1.3. The proof of the functional equation. The Gamma-function
plays an important part in the theory of the zeta-function (see [199], §1.86
and §4.41, for a collection of its most important properties). For Re z > 0,
the Gamma-function may be defined by Euler’s integral
Z ∞
Γ(z) =
uz−1 exp(−u) du.
0
2
Substituting u = πn x leads to
Z ∞
s s 1
s
−2
−1
2
π
=
x
exp(−πn2 x) dx.
(3.4)
Γ
s
2
n
0
Summing up over all n ∈ N yields
∞
∞ Z ∞
s X
X
s
1
− 2s
−1
2
=
x
π Γ
exp(−πn2 x) dx.
2 n=1 ns
n=1 0
On the left-hand side we find the Dirichlet series defining ζ(s); in view of its
convergence, the latter formula is valid only for σ > 1. On the right-hand
side we may interchange summation and integration, justified by absolute
convergence. Thus we obtain
Z ∞
∞
s
X
s
−1
− 2s
ζ(s) =
x2
exp(−πn2 x) dx.
π Γ
2
0
n=1
We split the integral at x = 1 and get
Z 1 Z ∞ s
s
− 2s
ζ(s) =
+
x 2 −1 ω(x) dx,
(3.5)
π Γ
2
0
1
where the series ω(x) is given in terms of the theta-function:
ω(x) :=
∞
X
n=1
exp(−πn2 x) =
1
(θ(x) − 1)
2
(since exp(−πn2 x) = exp(−π(−n)2 x) for any n ∈ N). In view of the functional equation for the theta-function,
√
1
1
1 √
1
=
θ
− 1 = xω(x) + ( x − 1),
ω
x
2
x
2
Chapter 3
110
Hecke theory
we find by the substitution x 7→ x1 that the first integral in (3.5) is equal to
Z ∞
Z ∞
s+1
1
1
1
− 2s −1
x
dx =
x− 2 ω(x) dx +
− .
ω
x
s−1 s
1
1
Substituting this in (3.5) yields
Z ∞
s
s+1
s
1
− 2s
ζ(s) =
+
x− 2 + x 2 −1 ω(x) dx.
(3.6)
π Γ
2
s(s − 1)
1
Since ω(x) ≪ exp(−πx), the last integral converges for all values of s, and
thus (3.6) holds, by analytic continuation, throughout the complex plane.
The right-hand side remains unchanged by s 7→ 1 − s. This proves the
functional equation for zeta. •
3.1.4. The case of Dirichlet L-functions. In a similar manner as
above one can prove the functional equation for Dirichlet L-functions L(s, χ)
with a primitive character χ mod q. Here we have to distinguish once again
the cases χ(−1) = +1 and χ(−1) = −1. In the first case we find
X
1
τ (χ)
2
,χ ,
(3.7)
θ(x, χ) :=
χ(n) exp(−πn x/q) = √ θ
qx
x
n∈Z
where the Gaussian sum τ (χ) is given by (1.26) and satisfies
(3.8)
τ (χ)τ (χ) = χ(−1)|τ (χ)|2 = χ(−1)q;
this formula being valid for any primitive character χ mod q. The second
case, χ(−1) = −1, is slightly more difficult. Here we make use of
X
τ (χ)
1
2
χ(n)n exp(−πn x/q) = √ 3 θ̃
(3.9)
θ̃(x, χ) :=
,χ ,
x
2
i
qx
n∈Z
The proofs of these functional equations rely on the Poisson summation formula (3.2) and basic facts about primitive characters. Formulae (3.7) and
(3.9) lead by more or less the same method as for the zeta-function to
Theorem 3.4. Let χ be a primitive character mod q. Then, for any s ∈ C,
q s+δ
τ (χ) q 1+δ−s
s+δ
1+δ−s
2
2
L(s, χ) = δ √
L(1 − s, χ),
Γ
Γ
π
2
i q π
2
where δ := 21 (1 − χ(−1)).
The Davenport-Heilbronn zeta-function is given by
1 + iκ
1 − iκ
L(s, χ) +
L(s, χ),
L(s) =
2
2
where
p
√
10 − 2 5 − 2
√
κ :=
5−1
Section 3.1
The functional equation
111
and χ is the character mod 5 with χ(2) = i. It is an easy consequence of Theorem 3.4 that the Davenport-Heilbronn zeta-function satisfies the functional
equation
1−s
2s s
s+1
5 2 5
L(s) =
L(1 − s).
Γ
Γ 1−
π
2
π
2
Davenport & Heilbronn [43] introduced this function as an example for a
Dirichlet series having infinitely many zeros on the critical line and also infinitely many zeros in the half-plane σ > 1 in spite of satisfying a Riemann-type
functional equation. The localization of these zeros is not too easy (see also
[200]). However, following Balanzario [11] we can give another examples:
consider of a Dirichlet series satisfying a Riemann-type functional equation
for which the analogue of the Riemann hypothesis does not hold. Consider
the following functions with 5-periodic Dirichlet coeffcients:
√
1
1
1
1
1+ 5
−s
(1 + 5 2 )ζ(s) = 1 + s + s + s +
+ ...,
2
3
4
5s
1
1
1
0
L(s, χ) = 1 − s − s + s + s + . . . ,
2
3
4
5
where χ is the character mod 5 with χ(2) = −1. Both functions satisfy the
same functional equation,
πs
1
(3.10)
F (s) = 5 2 −s 2(2π)s−1Γ(1 − s) sin F (1 − s)
2
(see (2.6) and (2.7)). Now let z be any complex number, then the Dirichlet
series
1
1
L(z, χ)(1 + 5 2 −s )ζ(s) − L(s, χ)(1 + 5 2 −z )ζ(z)
vanishes for s = z, satisfies the functional equation (3.10), has for σ > 1
a Dirichlet series expansion, and, obviously, this function is not identically
vanishing. Clearly, this example can easily be generalized (see [11]). We keep
in mind that a functional equation is not sufficient for having all complex zeros
on a straight line! Is it the Euler product which forces the nontrivial to lie
on the critical line?
An alternative approach toward the functional equation for Dirichlet Lfunctions uses another interesting class of Dirichlet series which do not have
an Euler product in general. In the following section we shall briefly introduce
3.1.5. Hurwitz zeta-functions. For σ > 1, the Hurwitz zeta-function
is given by
∞
X
1
,
ζ(s, α) =
s
(m
+
α)
m=0
where α is a parameter from the interval (0, 1]. The Hurwitz zeta-function
can be continued analytically to the whole complex plane except for a simple
112
Chapter 3
Hecke theory
pole at s = 1 with residue 1. Also these Dirichlet series satisfy some kind of
functional equation. One can show that
(∞
)
∞
X
X
πs
sin 2πnα
cos 2πnα
2Γ(1 − s)
+ cos
;
(3.11)
ζ(s, α) =
1−s
1−s
(2π)
n
2 n=1 n1−s
n=1
this formula is valid for σ < 0 (in view of the infinite series on the right-hand
side).
If α is rational, α = aq with coprime a and q, say, then we have
qs X
a
=
χ(a)L(s, χ)
(3.12)
ζ s,
q
ϕ(q) χ mod q
and
(3.13)
q
1 X
a
L(s, χ) = s
χ(a)ζ s,
.
q a=1
q
From these identities one can deduce the functional equation for Dirichlet
L-functions from the one for Hurwitz zeta-functions and vice versa (if the
parameter α is rational).
Hurwitz zeta-functions are of special interest with respect to Riemann’s hypothesis. For α ∈ { 21 , 1} the Hurwitz zeta-function is related to the Riemann
zeta-function:
1
= (2s − 1)ζ(s).
(3.14)
ζ(s, 1) = ζ(s)
and
ζ s,
2
However, besides α = 21 , 1 there are no identities of this type; more precisely,
in Steuding [190] it was proved that ζ(s, α)/ζ(s) is entire if and only if
α = 12 or 1. The distribution of zeros of ζ(s, α) as a function of s depends
drastically on the parameter α and this is very interesting as we shall briefly
explain. For instance, the Hurwitz zeta-function given by (3.14) vanishes
, k ∈ Z, and all other non-real zeros are expected to lie on the
for s = 2πik
log 2
critical line σ = 21 (by RH). This example is somehow special. It is known
that for any 21 < σ1 < σ2 < 1 and any transcendental or rational α 6= 12 , 1 the
function ζ(s, α) has more than cT zeros in the rectangle σ1 ≤ σ ≤ σ2 , |t| ≤ T ,
where c is a positive constant depending on σ1 , σ2 and α (actually, this is
a consequence of the universality property for the Hurwitz zeta-function;
see Garunkštis & Laurinčikas [54] or Karatsuba & Voronin [112]). This
behaviour is also expected to be true for algebraic irrational α (see Garunkštis
[53]). Denote by ρα = βα +iγα the nontrivial zeros of ζ(s, α) (these nontrivial
zeros are defined in a similar way as the ones for ζ(s); for short: apart
from finitely many exceptions they have a non-negative real part). However,
Section 3.1
The functional equation
113
Garunkštis & Steuding [55] (see also [54]) showed that
1
2π X
βα −
= log α.
lim
T →∞ T
2
|γα |≤T
Thus, the nontrivial zeros of Hurwitz zeta-function weighted by their distance
from σ = 21 have a tendency to lie to the left of the critical line and so any
reasonable analogue of the Riemann Hypothesis for ζ(s, α) fails for generic
α 6= 21 , 1.
y
110
100
90
1
0.8
a
0.6
1
0.5
0
x
Figure 1. Trajectories of several zeros of ζ(s, α), 1/2 ≤ α ≤
1; the 30-th zero of ζ(s) = ζ(s, 1) is plotted in green, the 35-th
in pink.
Now we want to study the zeros of ζ(s, α) as a function of α. By partial
summation,
1−s
Z ∞ 1
− {u}
3
1
1
1
2
+
+α
du,
+s
ζ(s, α) = s +
s
3
α
(1 + α)
s−1 2
(u + α)s+1
2
valid for σ > 0, where {u} denotes the fractional part of a real number u
(see Karatsuba & Voronin [112]). The integral converges uniformly for s
from any compact subset of the half-plane σ > 0 and arbitrary α. Hence,
ζ(s, α) is a continuous function in the variable s 6= 1 and the parameter
α and, in particular, the zeros depend continuously on α. Now lets assume
Riemann’s hypothesis for a short while and follow some idea from Garunkštis
& Steuding [56]. By (3.14), for any T and any δ > 0, there exists a positive
constant c = c(T, δ) such that all nontrivial zeros ρα = βα +iγα of all Hurwitz
zeta-functions ζ(s, α) with | 21 − α| ≤ c, which have imaginary part |γα| ≤ T ,
Chapter 3
114
Hecke theory
satisfy either |βα − 21 | ≤ δ or |βα − 0| ≤ δ. This scenario is illustrated in
Figure 1.1
3.1.6. Further proofs of the functional equation. Riemann also
gave a second proof of the functional equation by using. The starting point
is (3.4). From this formula we easily deduce, for σ > 1,
Z ∞
∞
∞
X
X
1
s−1
Γ(s)
=
exp(−nx) dx.
x
s
n
0
n=1
n=1
The sum on the right-hand side is a geometric series and thus we arrive at
the integral representation
Z ∞
1
xs−1
dx.
(3.15)
ζ(s) =
Γ(s) 0 exp(x) − 1
From this one can derive the formula
Z
exp(−πis)Γ(1 − s)
z s−1
ζ(s) =
dz,
2πi
C exp(z) − 1
where C is the contour which starts ay infinity on the positive real axis,
encircles the origen once in the positive direction, excluding the points s =
±2πi, ±4πi, . . ., and then returns to infinity. The sum of the residues at the
points s = ±2πin for n ∈ N is
∞
πs
πs X
(2πn)s−1 = 4πi exp(πis) sin (2π)s−1 ζ(1 − s)
−4πi exp(πis) sin
2 n=1
2
and this is already half of the proof (see (2.6) and (2.7)). This approach
is related to the proof of the approximate functional equation 2.8. It also
applies to Hurwitz zeta-functions and their functional equation as well to
studies on the values of the zeta-function at the negative integers (see the
following section).
Many further proofs of the functional equation were discovered; some of
them can be found in Titchmarsh [200]. However, there is one which has
to be mentioned explicitly since this approach has found several important
applications and generalizations, in particular, in algebraic number theory.
In his doctoral thesis from 1950 (see also [197]), Tate started to apply
harmonic analysis to local fields (in particular, Poisson’s summation formula).
He introduced integration techniques on the ring of ideles of a number field
and succeeded in isolating and identifying the contributions to the functional
equation from each of the ramified prime ideals. In the simplest case his
method gives a proof of the functional equation for the Riemann zeta-function
1We
would like to thank Michael Trott for the MATHEMATICA notebook for
M. Trott, Zeros of the Generalized Riemann Zeta Function as a Function of
a, Background image in graphics gallery, in Wolfram [216]; see also the webpage
http://documents.wolfram.com/v4/MainBook/G.2.22.html.
Section 3.1
The functional equation
115
which uses only local information. Recall that number fields and function
fields of curves over finite fields are called global fields and the completions
of a global field with discrete valuation and finite residue field are said to
be local; for instance, the p-adic fields constructed from the field of rational
numbers Q and R are local. The local fields contain deep information of the
underlying global field. For example, Hasse [82] proved that a quadratic form
with rational coefficients represents a given number over the global field Q if
and only if it does in all local fields Qp , p ≤ ∞, i.e, the p-adic fields Qp for
each prime p and the field of real numbers R = Q∞ (this notation is standard
in the theory of valuations). The so-called local-global principle is the idea
of putting together information from all local fields to get information in the
corresponding global field. Roughly speaking, Tate has given a dissection of
the functional equation into a family of local functional equations for each
p ≤ ∞ corresponding to the Euler factors for each prime p in the Euler
product for ζ(s) in addition with the contribution for the infinite prime, that
is the Gamma-factor. However, Tate’s method gives more; for example, the
easiest proof for the functional equation for Dedekind zeta-functions. For
more details on Tate’s thesis and its generalizations we refer to Tate [197]
and Swinnerton-Dyer [195].
3.1.7. The Phragmén-Lindelöf principle. Functional equations of
the Riemann-type contain important information on the order of growth.
In order to deduce this information we shall use a kind of maximum principle
for unbounded regions, the theorem of Phragmén-Lindelöf:
Lemma 3.5. Let f (s) be analytic in the strip σ1 ≤ σ ≤ σ2 with f (s) ≪
exp(ǫ|t|). If
f (σ1 + it) ≪ |t|c1
and
f (σ2 + it) ≪ |t|c2 ,
then f (s) ≪ |t|c(σ) uniformly in σ1 ≤ σ ≤ σ2 , where c(σ) is linear with
c(σ1 ) = c1 and c(σ2 ) = c2 .
A proof can be found in the paper of Phragmén & Lindelöf [135] or, for
example, in Titchmarsh [199]. Note that there are counterexamples if the
growth condition f (s) ≪ exp(ǫ|t|) is not fulfilled.
We illustrate the so-called Phragmén-Lindelöf principle with an easy application to the zeta-function. We define
µ(σ) = lim sup
t→∞
log |ζ(σ + it)|
.
log t
One can show that µ(σ) is a convex function of σ. Taking into account the
absolute convergence of the defining Dirichlet series we immediately see that
µ(σ) = 0 for σ > 1. The order of growth in the half-plane left of the critical
116
Chapter 3
Hecke theory
strip is ruled by the functional equation which we may rewrite as
ζ(s) = ∆(s)ζ(1 − s),
where ∆(s) is given by (2.7). Applying Stirling’s formula (1.53) we get, for
t ≥ 1,
12 −σ
t
|∆(σ + it)| ∼
,
2π
uniformly in σ. From this we deduce that
1
ζ(σ + it) ≍ |t| 2 −σ |ζ(1 − σ + it)|,
uniformly in σ, as |t| → ∞. This estimate implies now µ(σ) = 21 − σ for
σ < 0. For the calculation of µ(σ) with 0 ≤ σ ≤ 1 we apply the theorem of
Phragmén-Lindelöf, Lemma 3.5. It follows that µ(σ) is non-increasing and
convex downwards and we obtain explicit estimates from the estimates of
µ(σ) for σ outside of the critical strip. Altogether, we obtain

0
if
σ > 1,

1
µ(σ) ≤
(1 − σ) if 0 ≤ σ ≤ 1,
 21
−σ
if
σ < 0.
2
In view of the functional equation, resp. the convexity of µ(σ), the value for
σ = 21 is essential. In particular, we obtain µ( 21 ) ≤ 14 or, equivalently,
1
1
+ it ≪ t 4 +ǫ
ζ
2
as |t| → ∞, valid for any positive ǫ. Recall that the approximate functional
equation gave only the exponent 12 (see (2.8)). However, this is not the
best estimate for the zeta-function on the critical line. The exponent 41 is
called the convexity bound and there is a long list of improvements. At
32
+ ǫ. This
the moment, Huxley [94] holds the record with the exponent 205
remarkable estimate was obtained with a different method (namely, estimates
for exponential sums) but is still far away from the exponent ǫ predicted by
the Lindelöf hypothesis.
There are more advanced applications of the Phragmén-Lindelöf principle; however, often with respect to the modulus of some characters or other
arithmetic objects. Recently, new methods for breaking the corresponding
convexity bounds in these arithmetic cases as well as unexpected applications
were found; see Iwaniec & Sarnak [100].
Integrals like (3.5) are called Mellin transforms (and we already met them in
Chapter 1.6.7). Here we want to derive the Mellin inversion formula (3.16). If g(s)
Section 3.2
The zeta-function at the integers
117
is analytic in some right half-plane, then its inverse Mellin transform is given by
Z σ+i∞
1
g(s)x−s ds
f (x) =
2πi σ−i∞
for positive values of x such that the integral converges absolutely. By contour
integration, it turns out that the integral is independent of σ.
Exercise 72. Show that
Z ∞
(3.16)
g(s) =
f (x)xs−1 dx
0
⇐⇒
1
f (x) =
2πi
Z
σ+i∞
g(s)x−s ds.
σ−i∞
Hint: let x = exp(z), s = σ + 2πiy and rewrite the integrals according g(s) =
gσ (y), f (x) = fσ (z) exp(−σz); apply Fourier analysis.
Exercise 73. i) Show (3.8).
ii) Prove the identities (3.7) and (3.9).
Hint: for the second formula one may first prove
X
3 X
(n + α) exp(−π(n + α)2 /x) = −ix 2
n exp(−πxn2 + 2πinα).
n∈Z
n∈Z
iii) Deduce Theorem 3.4.
Exercise 74. * i) Prove identity (3.11). Start with the identity
Z s−1
exp((1 − α)z)
z
1
dz,
ζ(s, α) = exp(−πis)Γ(1 − s)
2πi C
exp(z) − 1
where C is the positively oriented contour consisting of the positive real part of the
axis from +∞ to 0, enclosing the point z = 0 by a circle of radius r ∈ (0, 2π), and
returning to +∞.
Hint: consult Garunkštis & Laurinčikas [54].
ii) Show the identities (3.12) and (3.13) and deduce from this representation and
(3.11) the functional equation for Dirichlet L-functions.
Exercise 75. Use the Phragmén-Lindelöf principle to prove estimates for the order
of growth of Dirichlet L-functions and Hurwitz zeta-functions. What can you do
for Dedekind zeta-functions?
3.2. The zeta-function at the integers
It is remarkable that already Euler [51] had partial results toward the functional equation for ζ(s), namely, formulae for the values of ζ(s) for integral
s and for half-integral s relating s with 1 − s (see Ayoub [5]). Here we want
to sketch his contribution briefly.
118
Chapter 3
Hecke theory
3.2.1. The positive integers. A famous problem in the 17/18th cenP
−2
tury was the evaluation of ζ(2) = ∞
n=1 n . This was solved by Euler in
1737 as follows: Comparing the product
∞ ∞ Y
z2
z Y
sin z
=
1− 2 2
=
1−
(3.17)
z
πn
π n
n=−∞
n=1
n6=0
with the power series representation
∞
one obtains
X
sin z
z2 z4
z 2k
=1−
+
∓ ... =
(−1)k
,
z
3!
5!
(2k + 1)!
k=0
∞
X
π2
1
= .
ζ(2) =
n2
6
n=1
Euler’s proof was much discussed by his contemporaries. First of all, it was
not clear whether sin z has no complex zeros; furthermore, the convergence of
(3.17) cannot be proved without complex analysis which was not developed
in those times. However, today Euler’s argument is waterproof and might be
the easiest proof of all.
As Euler we want to compute more values of the zeta-function at the integers. For this purpose we have to introduce the Bernoulli numbers (introduced by the Bernoullis, they are extraordinarily important in algebraic
number theory). The numbers Bn are defined by the identity
∞
X
zn
1
1
z
(3.18)
=
Bn = 1 − z + z 2 ∓ . . . .
exp z − 1 n=0
n!
2
12
The function expzz−1 + 2z is an even function. This and (3.18) imply that
B2n+1 = 0 for n ∈ N. Hence, one finds
∞
X
(2π)2k
d
sin(πz)
(−1)k
B2k z 2k = πz cot(πz) − 1 = z
log
.
(2k)!
dz
πz
k=1
Using the product representation (3.17), we find
∞
∞
X
X
d
z2
z
log 1 − 2 = −2
ζ(2k)z 2k .
dz
n
n=1
k=1
Comparing the coefficients, we arrive at
Theorem 3.6. For k ∈ N,
ζ(2k) = (−1)
k−1 (2π)
2k
2(2k)!
B2k .
Nearly nothing is known about the values of zeta at the positive odd integers;
in 1979, Apéry [1] proved that ζ(3) is irrational but the arithmetic character
of ζ(5) is still unknown.
Section 3.2
The zeta-function at the integers
119
3.2.2. The negative integers. Now we study the values at the negative
integers. Here Euler found
Theorem 3.7. For n ∈ N,
ζ(0) = − 21
and
ζ(−n) = −
Bn+1
.
n+1
It is remarkable that Euler found this formula since he considered ζ(s) as a
function of a real variable s and for that purpose he had to pass behind the
pole at s = 1 (which is in principle only possible by analytic continuation
leaving the real axis). Euler’s argument was as follows: we have, for m ∈ N0 ,
(3.19)
1m − 2m + 3m ∓ . . . = (1 − 2m+1 )ζ(−m),
and
m
x
d
.
x − 2 x + 3 x ∓... = x
dx
1+x
m
m 2
m 3
Using the latter formula with x = exp(2πiw) we get
(1 − 2m+1 )ζ(−m) = (2πi)−m
d
dw
m
exp(2πiw) 1 + exp(2πiw) .
w=0
By (3.18) this leads to the formula of Theorem 3.7. Euler’s proof needs a
modified notion of convergence – this is obvious with respect to (3.19); using
summability arguments one can make his approach waterproof. Here we shall
use an idea of Riemann to prove Theorem 3.7.
Proof. Let σ > 1. We start with the integral (3.15) and deduce via (3.18)
Z ∞
z s−1
dz
Γ(s)ζ(s) =
exp(z) − 1
0
Z ∞
n
X
zk
k
s−2
(−1) Bk dz
=
z exp(−z)
(3.20)
k!
0
k=0
Z ∞
∞
X
zk
k
s−2
(−1) Bk dz.
+
z exp(−z)
k!
0
k=n+1
The second integral is bounded by ≪ Γ(σ + n) and thus convergent and
analytic for σ > −n. The first integral is equal to
n
X Bk
1
Γ(s + k − 1);
Γ(s − 1) + Γ(s) +
2
k!
k=2
120
Chapter 3
Hecke theory
hence it is meromorphic in the whole of C. By the functional equation of the
Gamma-function, we deduce another analytic continuation of ζ(s) to σ > −n:
n
1 X Bk
1
(3.21)
+ +
s(s + 1) · . . . · (s + k − 2)
ζ(s) =
s−1 2
k!
k=2
Z ∞
∞
X
1
zk
s−2
+
z exp(−z)
(−1)k Bk dz.
Γ(s) 0
k!
k=n+1
In particular, we find at the poles of Γ(s)
n
1 1 X Bk
(1 − n)(−n) · . . . · (k − n − 1)
ζ(1 − n) = − + +
n 2
k!
k=2
n
1 X n
Bk .
= −
n k=0 k
By the recursion formula for the Bernoulli numbers (an exercise left for the
reader), this implies Euler’s formula. •
Euler [51] was aware about the correspondence between the values of ζ(2k)
and ζ(1 − 2k) for integers k:
ζ(2k) = (−1)k−1
(2π)2k
(2π)2k
B2k =
ζ(1 − 2k);
2(2k)!
2 cos πkΓ(2k)
this is indeed the functional equation in the form (1.58) for s = 2k. For more
details on Euler’s work on the zeta-function we refer to Ayoub [5].
3.2.3. A p-adic zeta-function. The value-distribution of the zetafunction for integer values allows the construction of a p-adic zeta-function
ζp (s) which interpolates ζ(s) (as a matter of fact, all values ζ(1−n) for n ∈ N0
are rational). This was first observed by Kubota & Leopoldt [122]; their construction implies remarkable and surprising facts on Bernoulli numbers, e.g.,
the old von Staudt-Clausen congruences
X
1
∈ Z,
Bm +
p
p
m≡0 mod (p−1)
valid for any positive even integer m, and the Kummer congruences
Bm
Bn
≡
mod p
if m ≡ n 6≡ 0 mod (p − 1).
m
n
The approximation by a p-adic zeta-function is as follows:
ζp (1 − m) = ζ(1 − m)(1 − pm−1 )
if m ≡ 0 mod p − 1;
it should be noticed that the factor 1 − pm−1 on the right-hand side is exactly
the Euler factor of ζ(1 −m) at p. Generalizations to Dirichlet L-functions are
important with respect to the p-adic analogue of the class number formula,
Section 3.3
Hamburger’s theorem
121
and elliptic analogues of the p-adic zeta-function are a major ingredient in
Wiles’ solution of Fermat’s last theorem. (We refer the interested reader to
Koblitz [116].)
Exercise 76. For n ∈ N, prove that B2n+1 = 0 and the recursion formula
n X
n
Bn = (−1)n
Bk .
k
k=0
Hint: first, show that
∞ X
n X
n
n=0 k=0
k
Bk
−z
zn
=
.
n!
exp(−z) − 1
Exercise 77. Using the differential equation cot′ (z) = −1 − cot z, prove the recursion formula
X
1
n+
ζ(2n) =
ζ(2k)ζ(2ℓ).
2
k+ℓ=n
−σ
Exercise 78. Let k, n ∈ N. Prove that ζ(s) = 1 +
√ O(2 n)nfor σ → 1∞,
and
1 + O n ) the
deduce (via Theorem 3.6 and Stirling’s formula n! = 2πn e
asymptotic formula
2k √
1
k
k−1
B2k = (−1) 4 πk
1+O
.
πe
k
Exercise 79. Prove that the probability that the probability that n randomly chosen
positive integers m1 , . . . , mn are coprime is equal to
Y
1
1
Prob (gcd(m1 , . . . , mn ) = 1) =
1− n =
.
p
ζ(n)
p
3.3. Hamburger’s theorem
In 1921, Hamburger [74] proved that the Riemann zeta-function is characterized by its functional equation.
Theorem 3.8. Let G(s) be an entire function of finite order, P (s) a polynomial, and suppose that
∞
G(s) X a(n)
=
,
f (s) :=
P (s) n=1 ns
the series being absolutely convergent for σ > 1. Assume that
s
1−s
− 1−s
− 2s
f (s) = π 2 Γ
g(1 − s),
(3.22)
π Γ
2
2
Chapter 3
122
Hecke theory
where
∞
X
b(n)
,
g(1 − s) =
n1−s
n=1
the series being absolutely convergent for σ < −α for some positive constant
α. Then f (s) = cζ(s), where c is a constant.
We shall give here a simplified proof due to Siegel [181].
Proof. By (3.4) we find, for x > 0,
Z 2+i∞
s
s
1
(πx)− 2 ds
f (s)Γ
φ(x) :=
2πi 2−i∞
2
Z
∞
2+i∞ X
s
s
1
(πn2 x)− 2 ds
Γ
=
a(n)
2πi 2−i∞
2
n=1
(3.23)
= 2
∞
X
a(n) exp(−πn2 x).
n=1
In view of (3.22) we also have
1
φ(x) =
2πi
Z
2+i∞
2−i∞
g(1 − s)Γ
s
s−1
1−s
π 2 x− 2 ds.
2
Next we move the line of integration from the line σ = 2 to σ = −1 − α.
Obviously, f (s) is bounded on σ = 2 and g(1 − s) is bounded on σ = −1 − α.
By Stirling’s formula (1.53),
Γ 2s
1
≪ |t|σ− 2
1−s
Γ 2
3
as |t| → ∞. Thus, g(1 − s) ≪ |t| 2 on σ = 2 as |t| → ∞, and, justified by
the Phragmén-Lindelöf principle (see Lemma 3.5), we can apply Cauchy’s
theorem. It follows that
Z −1−α+i∞
k
X
s−1
s
1−s
1
−
π 2 x 2 ds +
g(1 − s)Γ
Rj ,
(3.24)
φ(x) =
2πi −1−α−i∞
2
j=1
where R1 , . . . , Rk are the residues at the poles, say s1 , . . . , sk . It is easily seen
that the sum of residues is of the form
k
X
j=1
Rj =
k
X
j=1
sj
x− 2 Pj (log x) =: R(x),
Section 3.3
Hamburger’s theorem
123
where the Pj (log x) are polynomials in log x. We rewrite (3.24) and find as
above
2 s−1
Z −1−α+i∞ ∞
2
1
πn
1−s
1 X
b(n)
Γ
φ(x) = √
ds + R(x)
x n=1
2πi −1−α−i∞
2
x
∞
2 X
b(n) exp(−πn2 /x) + R(x).
= √
x n=1
Comparing with (3.23), we arrive at
∞
X
∞
1 X
1
b(n) exp(−πn2 /x).
a(n) exp(−πn x) − R(x) = √
2
x
n=1
n=1
2
Multiplying with exp(−πt2 x) with t > 0 and integrating over (0, ∞) with
respect to x, we get
Z
∞
∞
X
t ∞
t X a(n)
2
R(x)
exp(−πt
x)
dx
=
b(n) exp(−2πnt).
−
π n=1 (t2 + n2 ) 2 0
n=1
The integral can be evaluated as a finite sum of terms of the form
Z ∞
Q(t; a, b) :=
xa (log x)b exp(−πt2 x) dx,
0
where the b’s are integers and Re a > −1; thus, Q(t; a, b) is a sum of terms
of the form tα (log t)β . Hence,
∞
∞
X
X
π
1
1
− t Q(t; a, b) = π
−
b(n) exp(−2πnt).
a(n)
t
−
in
t
+
in
2
n=1
n=1
The left-hand side is a meromorphic function in t with poles at t = ±in
for n ∈ N. The right-hand side is periodic with period i and, by analytic
continuation, the function on the left-hand side is also periodic. Hence, the
residues at in and i(n + 1) are equal. Thus, a(n) = a(n + 1) for all n ∈ N
and Hamburger’s theorem is proved. •
As Hecke pointed out in 1936, this result is better understood in the context
of modular forms. However, before we start with Hecke’s theory we have to
recall some basic facts about modular forms.
Exercise 80. * Try to find an analogue of Hamburger’s theorem for Dirichlet Lfunctions! Note that there are many L(s, χ) satisfying the same functional equation;
what do they have in common?
Chapter 3
124
Hecke theory
3.4. Modular forms
Modular forms are holomorphic functions of the upper half-plane which are
almost invariant under operations of the modular group (resp. subgroups).
In the recent past they have been proven to be of greatest importance in
modern number theory, e.g., in Wiles’ proof of Fermat’s last theorem. For
the details of the theory we refer to Iwaniec [99], Koblitz [117], and Miyake
[146].
3.4.1. Eisenstein series and the discriminant. Recall that the set
of all 2 × 2 matrices with integral entries and determinant 1 forms a group,
the so-called special linear group over Z, denoted by SL2 (Z). This group is
generated by the two matrices
0
1
1 1
.
and
−1 0
0 1
In the sequel we shall study the transformation properties of holomorphic
functions of the upper half-plane
H := {z ∈ C : Im z > 0}
under the action of SL2 (Z)-matrices as fractional linear transformations
a b
az + b
∈ SL2 (Z).
for M :=
z 7→ Mz :=
c d
cz + d
We start with an example.
For z ∈ H and a fixed positive even integer k > 2, the Eisenstein series of
weight k is defined by
Gk (z) =
(k − 1)!
2(2πi)k
X
m,n∈Z
(m,n)6=(0,0)
1
(mz + n)k
(the condition k > 2 is needed to guarantee absolute convergence). What
happens with Gk (z) under transformations of the special linear group? The
action of M = ( ac db ) ∈ SL2 (Z) on this function replaces (m, n) by (am +
cn, bm + dn) and therefore permutes the terms of the sum. We obtain
az + b
= (cz + d)k Gk (z).
(3.25)
Gk
cz + d
We want to derive a more convenient expression for Gk (z). Recall the Lipschitz formula
(3.26)
X
n∈Z
∞
1
(−2πi)k X k−1
d exp(2πidz),
=
(z + n)k
(k − 1)! d=1
Section 3.4
Modular forms
125
which is valid for k ≥ 2 and z ∈ H. Using this formula, we find by splitting
the Gk -defining sum into terms with m = 0 and the terms with m 6= 0 that
!
∞
∞
X
(k − 1)! X
1
(k − 1)! X 1
+
Gk (z) =
(2πi)k n=1 nk m=1 (2πi)k
(mz + n)k
n∈Z
k
= (−1) 2
∞ X
∞
X
(k − 1)!
dk−1 exp(2πidmz).
ζ(k) +
(2π)k
m=1 d=1
In view of the values of the zeta-function at the integers, Theorems 3.6 and
3.7, we get the Fourier series expansion
(3.27)
∞
Bk X
Gk (z) = −
+
σk−1 (n) exp(2πinz);
2k n=1
here σk−1 (n) denotes the sum of divisors of n in the power k − 1. In fact,
this representation is the starting point for the approach of Bump & Beineke
[16] to power moments of ζ(s); the hope is that spectral theory for Eisenstein
series may be used to handle higher moments of the zeta-function.
A further example for the objects we want to study is the so-called discriminant, for z ∈ H, defined by
(2π)12
240G4(z))3 − (504G6 (z))2
1728
(the name discriminant comes from the theory of elliptic curves). In view of
(3.25) it follows that
(
3 2 )
az + b
(2π)12
az + b
az + b
∆
=
240G4
− 504G6
cz + d
1728
cz + d
cz + d
(3.28)
∆(z) =
(2π)12 (240G4(z))3 − (504G6 (z))2
1728
12
= (cz + d) ∆(z)
= (cz + d)12
for all M = ( ac db ) ∈ SL2 (Z). One can prove the following representation as
an infinite product:
12
∆(z) = (2π) exp(2πiz)
∞
Y
(1 − exp(2πinz))24
n=1
(a proof can be found in Koblitz [117]). The Fourier series expansion takes
the form
∞
X
12
τ (n) exp(2πinz)
∆(z) = (2π)
n=1
Ramanujan [173] conjectured that the coefficients τ (n) are multiplicative
11
and satisfy the estimate |τ (p)| ≤ 2p 2 . The multiplicativity was proved by
126
Chapter 3
Hecke theory
Mordell [150], in particular by the beautiful formula
mn X
.
τ (m)τ (n) =
d11 τ
d2
d|(m,n)
The estimate was shown by Deligne in a more general setting (see (3.32)
below).
3.4.2. Definitions and basic facts. The functions Gk (z) and ∆(z)
have remarkable transformation properties with respect to SL2 (Z). They are
examples of modular forms to the full modular group. Here comes the general
definition.
The group Γ := SL2 (Z) is called the modular group. We will also consider
subgroups. For a non-negative integer k and a positive integer N, we define
a b
∈ SL2 (Z) : c ≡ 0 mod N ;
Γ0 (N) =
c d
clearly, this defines a subgroup of the full modular group Γ = Γ0 (1) and is
called Hecke subgroup of level N or congruence subgroup mod N.
A holomorphic function f on H is said to be a modular form of weight k
for Γ0 (N) if
a b
az + b
k
∈ Γ0 (N),
= (cz + d) f (z)
for all
(3.29)
f
c d
cz + d
and f (z) is holomorphic at infinity, i.e., f (z) has a Fourier series expansion
(3.30)
f (z) =
∞
X
a(n) exp(2πinz).
n=0
A modular form f is said to be a cusp form if f vanishes at all cusps or,
equivalently, if
z = x + iy 7→ y k |f (z)|2
is bounded on H. Then we have a(0) = 0 in the Fourier expansion (3.30) for
f (z).
The modular forms on Γ0 (N) of weight k form a finite dimensional complex
vector space, denoted by Mk (Γ0 (N)); analogously, also the set of all cusp
forms on Γ0 (N) of weight k form is a finite dimensional complex vector space,
denoted by Sk (Γ0 (N)). For instance, the Eisenstein series Gk (z) defined by
(3.27) with k ≥ 4 are modular forms of weight k for the full modular group:
Gk ∈ Mk (Γ).
One can show that the space of all modular forms to the full modular
groups is the direct sum of all spaces Mk (Γ) with non-negative weights k,
Section 3.4
Modular forms
127
where Mk (Γ) has dimension

if k ≡ 2 mod 12,
 [k/12]
dim Mk (Γ) =
1 + [k/12] if k ≡ 0, 4, 6, 8, 10 mod 12,

0
otherwise;
the case of odd k follows immediately from the observation that any solution
of (3.29) with odd k vanishes identically. Moreover, one has the decomposition
Mk (Γ) = Gk C ⊕ Sk (Γ)
if k > 2, and every modular form for the full modular group is a polynomial
in the Eisenstein series G4 and G6 .
On the space of cusp forms one can introduce an inner product, the Petersson inner product, defined by
Z
dx dy
hf, gi :=
f (z)g(z)y k 2
y
H/Γ0 (N )
for f, g ∈ Sk (Γ0 (N)). Suppose that M|N. If f ∈ Sk (Γ0 (M)) and dM|N,
then z 7→ f (dz) is a cusp form on Γ0 (N) of weight k too. The forms which
may be obtained in this way from divisors M of the level N with M 6= N span
a subspace Sold
k (Γ0 (N)), called the space of oldforms. Its orthogonal complement with respect to the Petersson inner product is denoted Snew
k (Γ0 (N)).
For n ∈ N define the Hecke operator T (n) by
az + b
1 X k X
f
a
T (n)f :=
n ad=n 0≤b<d
d
for f ∈ Sk (Γ0 (N)). These operators are multiplicative and encode plenty
of arithmetic information of modular forms. The theory of Hecke operators
implies the existence of an orthogonal basis of Snew
k (Γ0 (N)) made of eigenvalues of the T (n) for n coprime with N. By the multiplicity-one principle
of Atkin & Lehner [4], the elements f of this basis are in fact eigenvalues of
all T (n), i.e., there exist complex numbers λf (n) for which T (n)f = λf (n)f
and a(n) = λf (n)a(1) for all n ∈ N. Furthermore, it follows that the first
Fourier coefficient a(1) of such an f is non-zero. A newform f is defined to
be an element of this basis normalized to have a(1) = 1. The newforms form
a finite set which is an orthogonal basis of the space Snew
k (Γ0 (N)).
To give an example, the discriminant ∆(z) given by (3.28) (sometimes also
called Ramanujan’s cusp form) is a cusp form of weight 12 for the full modular
group, and hence, after normalization, a newform of level 1.
3.4.3. Dirichlet series associated with modular forms. In the
1930’s Hecke [88] started investigations on modular forms and the associated
Chapter 3
128
Hecke theory
Dirichlet series (we already mentioned Hecke operators). Given a modular
form f with Fourier expansion (3.30), we may define the Dirichlet series
L(s, f ) =
(3.31)
∞
X
a(n)
n=1
ns
;
note that here the Fourier coefficient a(0) does not appear. By classic estimates for the Fourier coefficients of f this series converges in some half-plane
and its properties will be the main theme in the following section. However, for our later purpose we have to consider the case of a newforms more
detailed.
Suppose that f is a newform of weight k. In this case Deligne [45] proved
for the Fourier coefficients the estimate
|a(n)| ≤ n
(3.32)
k−1
2
d(n),
P
where d(n) = d|n 1 is the divisor function. In view of the classic bound
d(n) ≪ nǫ it follows that the series (3.31) converges absolutely for σ > k+1
.
2
By the multiplicativity of the Fourier coefficients it turns out that, in the
half-plane of absolute convergence, there is an Euler product representation
for the associated Dirichlet series:
−1 Y −1
Y
1
a(p)
a(p)
1 − s + 2s+1−k
.
L(s, f ) =
1− s
p
p
p
p∤N
p|N
Hecke [88] resp. Atkin & Lehner [4] (for newforms) proved that the Lfunction L(s, f ) has an analytic continuation to an entire function and satisfies the functional equation
s
N 2 (2π)−s Γ(s)L(s, f )
(3.33)
k
= ω(−1) 2 N
k−s
2
(2π)s−k Γ(k − s)L(k − s, f ),
where ω = ±1 is the Atkin-Lehner eigenvalue of the Atkin-Lehner involution
0 −N
on Sk (Γ0 (N)).
1 0
Exercise 81. * Prove the Lipschitz formula (3.26).
Hint: apply the Poisson summation formula (Theorem 3.2) to the function f (x) =
(x + iy)−k , where y is a positive real number and k ≥ 2 an integer.
Exercise 82. Show that Eisenstein series Gk (z) converge absolutely if k > 2.
Exercise 83. Prove that the j-function defined by
j(z) =
(240G3 (z))3
∆(z)
for
z∈H
is a modular function (i.e., a modular form of weight k = 0 to the full modular
group).
Section 3.5
Hecke’s converse theorem
129
3.5. Hecke’s converse theorem
In 1936, Hecke [88] proved a bijection between modular forms and Dirichlet series satisfying a Riemann-type functional equation; this includes Hamburger’s theorem as a special case. Moreover, it connects the theory of modular forms with the theory of Dirichlet series. In the sequel we follow Ogg’s
monograph [163].
3.5.1. The converse theorem. Let λ be
define the Hecke group G(λ) as the subgroup of
0
1 λ
,
G(λ) =
−1
0 1
a positive real number and
SL2 (R) given by
1
.
0
The case of the full modular group is Γ = G(1). Thus G(λ) is generated by
the fractional linear transformations
1
z 7→ z + λ
and
z 7→ − .
z
Extending the notion of modular forms of the modular group or its subgroups,
Hecke introduced G(λ)-modular forms as follows. A modular form of G(λ)
of weight k and multiplier ǫ ∈ {±1} is a holomorphic function f : H → C
satisfying
1
= ǫ(−iz)k f (z)
f (z + λ) = f (z)
and
f −
z
and having a Fourier expansion
∞
X
(3.34)
f (z) =
a(n) exp(2πinz/λ)
n=0
for all z ∈ H (this should be compared with (3.29)); this representation
includes the λ-periodicity and shows that f (z) is holomorphic at ∞. The
complex vector space of such modular forms which in addition satisfy the
growth condition a(n) = O(nc ) for some constant c is denoted by M0 (λ, k, ǫ).
A modular form of M0 (λ, k, ǫ) is a cusp form if a(0) = 0.
Hecke proved a one-to-one correspondence between the elements of
M0 (λ, k, ǫ) and Dirichlet series satisfying a Riemann-type functional equation plus some growth conditions. In fact, his theorem is even more general
since it also contains the case of functional equations relating two different
functions (Dirichlet series or modular forms):
Theorem 3.9. Let λ and k be fixed positive real numbers. Given two sequences {a(n)}n∈N0 and {b(n)}n∈N0 of complex numbers satisfying
a(n), b(n) ≪ nc
as
n→∞
Chapter 3
130
Hecke theory
for some positive constant c, we define
φ(s) =
∞
X
a(n)
n=1
as well as
Φ(s) =
λ
2π
s
ns
Γ(s)φ(s)
and
ψ(s) =
∞
X
b(n)
ns
n=1
and
Ψ(s) =
λ
2π
s
,
Γ(s)ψ(s).
Furthermore, let
f (z) =
∞
X
a(n) exp(2πinz/λ)
and
g(z) =
n=0
∞
X
b(n) exp(2πinz/λ).
n=0
Then the functions φ(s) and ψ(s) are analytic in the half-plane σ > c +
1, while f (z) and g(z) are analytic in the upper half-plane satisfying the
boundary condition
(3.35)
f (x + iy),
g(x + iy) ≪ y −c−1
as y → 0 + .
Furthermore, the following statements are equivalent:
(i) The function
a(0) ǫb(0)
+
,
s
k−s
is entire and bounded on every vertical strip and satisfies the functional equation
Φ(s) +
Φ(s) = ǫΨ(k − s);
(ii) For any z ∈ H,
k i
−1
f (z) = ǫ
.
g
z
z
Following Hecke we assign to any Dirichlet series φ(s) satisfying the conditions
of Theorem 3.9 its signature {λ, k, ǫ} by putting a(n) = b(n). Hecke’s theorem
includes the case of the zeta-function as φ(s) = ζ(2s), f being the thetafunction θ(x) and k = 21 (in particular, we see that the theta-function is a
modular form); the signature of zeta is {2, 12 , 1}.
3.5.2. Proof of the converse theorem. As in Riemann’s proof of the
functional equation or in Siegel’s proof of Hamburger’s theorem we shall
consider Mellin transforms and use the Mellin inversion formula (resp. the
Poisson summation formula) to prove the equivalence in Theorem 3.9.
However, first of all, we observe that the statement concerning the convergence of the Dirichlet series is trivial (by standard arguments as in the case of
ζ(s)). In order to derive the holomorphy and the boundary condition (3.35)
Section 3.5
Hecke’s converse theorem
131
for the Fourier series it suffices to consider the function f (z) only. Since, by
Stirling’s formula (1.53),
Γ(c + 1 + n)
−c − 1
n
∼ c1 nc
=
(−1)
Γ(c + 1)Γ(n + 1)
n
with some positive constant c1 , the Fourier series for f (x + iy) is dominated
term-by-term by
∞
X
−c − 1
n
exp(−2πny/λ) = (1 − exp(−2πy/λ))−(c+1)
(−1)
n
n=0
≪ y −c−1.
Conversely, given the boundary condition, we can bound the Fourier coefficients a(n) using their integral representation with y = n1 ,
Z 1 i
i
exp −2πin x +
/λ dy,
a(n) =
f x+
n
n
0
by O(nc exp(2π/λ)).
It remains to show the equivalence of (i) and (ii). We start with the implication (ii)⇒(i).
We note that, for sufficiently large σ,
Z ∞ s
∞
X
λ
xs−1 exp(−nx) dx
a(n)
Φ(s) =
2π
0
n=1
Z
∞
∞
X
=
a(n)y s−1 exp(−2πny/λ) dy
n=1
0
(as in the proof of the functional equation for ζ(s)). Now interchanging
summation and integration (justified by absolute convergence), we get
Z ∞X
∞
Φ(s) =
a(n)y s−1 exp(−2πny/λ) dy
0
=
Z
n=1
∞
0
y s−1(f (iy) − a(0)) dy.
The integral is improper for y → 0+ and y → ∞; we consider the contributions of the intervals (0, 1) and (1, ∞) separately. Since f (iy) − a(0) ≪
exp(−cy) as y → ∞ for some positive constant c, it follows that
Z ∞
y s−1(f (iy) − a(0)) dy
1
converges uniformly on vertical strips, and so it defines an entire function
which is bounded on vertical strips. For the integral taken over (0, 1) we
Chapter 3
132
Hecke theory
have to make use of (ii). We have
Z ∞
Z 1
i dy
a(0)y s 1
1−s
s−1
+
y f
.
y (f (iy) − a(0)) dy = −
s
y y2
y=0
1
0
Now by (ii) we get
Z 1
y s−1(f (iy) − a(0)) dy
0
Z ∞
a(0)
b(0)
= −
+ǫ
.
y k−s−1(g(iy) − b(0)) dy − ǫ
s
k−s
1
Hence,
a(0) ǫb(0)
+
Φ(s) +
s
k−s
Z ∞
s−1
=
y (f (iy) − a(0)) + ǫy k−s−1(g(iy) − b(0)) dy
1
is an entire function bounded on vertical strips. Furthermore, we observe
that (i) holds.
Now we assume (i) and deduce (ii). We shall use the formula
Z α+i∞
1
x−s Γ(s) ds,
(3.36)
exp(−x) =
2πi α−i∞
where α > 0 and x > 0; this is the Mellin inversion of Euler’s integral
representation of the Gamma-function. It follows that
Z α+i∞
1
(3.37)
f (iy) − a(0) =
y −s Φ(s) ds;
2πi α−i∞
however, here we have to choose the abscissa α > k such that the path of
integration lies inside the half-plane of absolute convergence for φ(s). We
shall move the path of integration over the origin to the left. Incorporating
the residues at s = 0 and s = k, we obtain
Z −α+i∞
1
f (iy) − a(0) =
y −s Φ(s) ds
2πi −α−i∞
+ {Res s=0 + Res s=k } y −s Φ(s).
(3.38)
In view of (i) we have
Res s=0 y −s Φ(s) = −a(0)
Res s=k y −s Φ(s) = ǫb(0)y −k .
and
Thus, we may replace (3.38) by
f (iy) − ǫb(0)y
−k
1
=
2πi
Z
−α+i∞
−α−i∞
y −s Φ(s) ds.
Section 3.5
Hecke’s converse theorem
133
Taking into account (i), we get
f (iy) − ǫb(0)y
−k
1
=
2πi
=
ǫ
2πi
Z
−α+i∞
y −sǫΨ(k − s) ds
−α−i∞
k+α+i∞
Z
y −(k−s)Ψ(s) ds,
k+α−i∞
by substituting s by k − s. The right-hand side above is equal to
i
−k
− b(0)
g
ǫy
y
(by the same argument as for (3.37)). This gives (ii) and the theorem is
proved. •.
3.5.3. The arithmetical and topological character of Hecke
groups. The groups G(λ) operate discontinuously as groups of fractional
linear transformations on H if and only if either λ > 2 or
π
λ = λm := 2 cos
with 3 ≤ m ∈ N ∪ {∞}.
m
The space M0 (λm , k, ǫ) with λm < 2 is non-trivial, i.e., 6= {0}, if and only if
ℓ
+ 1 − ǫ for some positive integer ℓ. In this case
k = 4 m−2
ℓ + (ǫ − 1)/2
.
dim M0 (λm , k, ǫ) = 1 +
m
The space of cusp forms is non-trivial if and only if dim M0 (λm , k, ǫ) ≥ 2; in
view of the dimension
√ √ formula this condition holds when k is suitably large.
For λm ∈ {1, 2, 3, 2} (i.e., m ∈ {3, 4, 6, ∞}), the Hecke group G(λm ) can
be defined arithmetically and in these cases G(λm ) holds a structure comparable to the full modular group Γ := G(1). We shall say a few words about
the properties of Dirichlet series associated with modular √
forms
√ to Hecke
groups G(λ). The situation is similar in the case of λm ∈ { 2, 3, 2} since
then the groups G(λm ) are conjugate to index 2 extensions of the congruence
subgroups Γ0 (N) of levels N = 2, 3, 4, respectively. However, in these cases
only the newforms have a basis consisting of normalized eigenfunctions (for
the Hecke and Atkin-Lehner operators). More details can be found by Hecke
[88], Atkin & Lehner [4], and the monographs of Ogg [163] and Miyake [146].
To indicate the difference between modular forms to Hecke groups G(λ)
which can be arithmetically defined and those who cannot, we state a result
of Wolfart.
/
√ √ In [215], he has shown that every space M0 (λm , k, ǫ) with λm ∈
{1, 2, 3, 2} has a basis consisting of modular forms of type
(3.39)
f (z) =
∞
X
n=0
r(n)an exp(2πinz/λ),
Chapter 3
134
Hecke theory
where r(n) ∈ Q and a is transcendental; moreover, a depends only on the
space M0 (λm , k, ǫ) and not on the modular form f . Clearly, the same statement holds for cusp forms.
Exercise 84. Show in detail that Hecke’s converse theorem 3.9 contains the case
of the zeta-function.
Exercise 85. Prove Formula (3.36) by i) using the Mellin inversion formula
(3.16), and ii) by the calculus of residues.
Hint: for ii), note that the sum of residues of the integrand is equal to
∞
X
Res s=−m x−s Γ(s) =
m=0
∞
X
(−x)m
.
m!
m=0
Exercise 86. What is the signature of a Dedekind zeta-function to an imaginary
quadratic number field? What can you deduce from Hecke’s theorem for general
Dedekind zeta-functions?
Exercise 87. Prove that Lf (s) with f being a basis element of the form (3.39)
has no Euler product representation.
3.6. Shimura-Taniyama-Wiles
The Shimura-Taniyama conjecture was first stated by Shimura & Taniyama
in 1955. Roughly speaking, it states that for any elliptic curve defined over
Q, there is a modular form such that both objects have the same L-function.
In the 1980’s, Frey observed that a counterexample to Fermat’s last theorem
would lead to a counterexample to the Shimura-Taniyama conjecture; this
was rigorously proved by Ribet soon after. The implication that the ShimuraTaniyama conjecture implies the truth of Fermat’s last theorem relates two
deep open conjectures from rather different fields. Fermat’s last theorem is
the famous claim of the 16-th century mathematician Fermat that all integer
solutions to the diophantine equation
Xn + Y n = Zn
with 3 ≤ n ∈ N
are trivial, i.e., xyz = 0; it refused its solution for more than 350 years unless
in 1995 Wiles [213] proved (in parts jointly with Taylor and building on
works of many others) an essential part of the Shimura-Taniyama conjecture,
namely, that any semistable elliptic curve (i.e., with squarefree conductor)
is modular. The full conjecture was proved by Breuil et al. [26]. Here we
want to motivate the link between geometry and number theory predicted by
the Shimura-Taniyama conjecture. For more details we refer to Knapp [115],
Section 3.6
Shimura-Taniyama-Wiles
135
Koblitz [117], and Washington [207]; for the amazing story behind Wiles’
proof read Singh [184].
3.6.1. Elliptic curves and their L-functions. An elliptic curve E
over some field K is a non-singular cubic curve f (X, Y ) = 0 with a K-rational
point (which may be a point at infinity). If char K 6= 2, 3, then the cubic can
be written as
(3.40)
Y 2 = X 3 + aX + b
with a, b ∈ K;
if the characteristic is 2 or 3, then slightly more complicated normal forms
have to be considered. However, for the sake of simplicity we may assume
that E is the set of K-solutions (x, y) to the diophantine equation above
plus the points at infinity. Further, we may assume that the elliptic curve is
defined over the rationals, i.e., a, b ∈ Q.
A famous theorem of Mordell [151] states that the set of rational points on
an elliptic curve forms a finitely generated abelian group. In particular, this
means that we can add points on an elliptic curve and their sum is a further
point on this curve. This fact relies on the simple observation that a generic
straight line has three intersection points with the cubic equation (3.40). This
gives an algebraic relation between any two given points P1 and P2 on the
elliptic curve and a third one, Q = (x, y) say. For some reasons we do not
explain, one cannot take Q to be the sum of P1 and P2 , but replacing Q by
its conjugate with respect to the x-axis is doing the job: P1 + P2 = (x, −y).
4
2
-2
-1
1
2
3
-2
-4
Figure 2. Adding points on an elliptic curve; here (−1, 0) +
(0, 1) = (2, −3) on the elliptic curve given by Y 2 = X 3 + 1.
One can show that this group of rational points has the form
(3.41)
E(Q) ∼
= T ⊕ Zr ,
where T is a finite group consisting of the torsion points and r is a nonnegative integer, called the rank of the elliptic curve E, which measures
Chapter 3
136
Hecke theory
the size of E. This structure of elliptic curves makes them a useful tool in
cryptography.
The construction of L-functions associated with elliptic curves is due to
Hasse [83] and his contemporaries. For this aim we have to study the reduction of E modulo the prime numbers. For prime p, denote by ν(p) the
number of solutions of (3.40) in Z/pZ, i.e., the number of solutions to the
congruence
Y 2 ≡ X 3 + aX + b mod p,
where the rational numbers a, b now have to be taken as the corresponding
residues modulo p (via the canonical projection onto Z/pZ). Define
λ(p) = p − ν(p) = p + 1 − ♯E(Z/pZ)
and put
−1 Y −1
Y
1
λ(p)
λ(p)
1 − s + 2s−1
,
LE (s) =
1− s
p
p
p
p|∆
p∤∆
where ∆ := −16(4a3 + 27b2 ) 6= 0 is the discriminant of E and which is
non-zero since E is by definition non-singular (the discriminant is intimately
related to the discriminant modular form ∆(z) defined by (3.28)). Hasse
proved that
√
(3.42)
|λ(p)| < 2 p;
this inequality might be regarded as the analogue of the Riemann hypothesis
for the (local) congruence zeta-functions (it is also related to Deligne’s estimate (3.32)). Consequently, the Euler product for LE (s) converges absolutely
for Re s > 23 . The analytic continuation of LE (s) to an entire function and
a Riemann-type functional equation were conjectured by Hasse and, apart
from partial cases, were proved only recently by the proof of the full ShimuraTaniyama conjecture by Wiles et al. [213, 26]. The functional equations has
the form
√ !s
√ !2−s
N
N
(3.43)
Γ(s)LE (s) = ±
Γ(2 − s)LE (2 − s)
2π
2π
where N is the conductor of the elliptic curve E, that is an integer built
from the prime divisors of the discriminant of E. Indeed, this reminds us of
the functional equation for Dirichlet series associated with modular forms of
weight k = 2.
One of the big yet unsolved questions is the Birch – Swinnerton-Dyer conjecture [17] (which is another millennium problem). In view of (3.42), the
number Np of points of an elliptic curve by reduction modulo a prime p lies
Section 3.6
Shimura-Taniyama-Wiles
137
in the interval
p+1−
√
p < Np := ♯E(Z/pZ) < p + 1 +
√
p.
It was shown by Lenstra [130] that the set of values which Np assumes in the
Hasse interval if one varies over all elliptic curves E mod p is quite similar
to the one of a random integer (an item which is of great importance for
their cryptographical use). We may expect that if the elliptic curve E has
infinitely many rational points, so if the rank r in (3.41) is positive, then
these points would be a rich source for many points by reduction modulo p
and Np would be large. On the contrary, if r = 0, then Np would straddle
both sides of p + 1 equally. We may rewrite (3.42) as λ(p) = 2Re αp with
√
αp = p exp(iθp ), θp ∈ R. In order to measure the relative size of the Np
with respect to p as p varies we may consider
Y
Y Np Y λ(p)
αp
p αp
1−
≈
1 − s + 2s =
1−
p
p
p
p
p
s=1
p
p
p∤∆
These infinite products do not converge; however, by the properties of the Lfunction associated with E, the right-hand side has an analytic continuation
and might be regarded as the reciprocal of the value LE (s) at s = 1. Now,
roughly speaking, the Birch – Swinnerton-Dyer conjecture claims that the
rank r of the Mordell group of an elliptic curve (3.41) is equal to the order
of vanishing of the associated L-function LE (s) at s = 1. Goldfeld [62]
examined that
√
Y Np
2
(log x)r ,
∼
p
L̃
(1)
E
p≤x
p∤∆
where L̃E (s) is the Euler product LE (s) restricted to those primes p which
do not divide the discriminant ∆. Furthermore, he showed that the Birch –
Swinnerton-Dyer conjecture implies the Riemann hypothesis for LE (s), i.e.,
the non-vanishing of LE (s) for σ > 1; this implication shows its deepness and
so we might be sceptic about a solution in the very near future. For more
information we refer to the survey of Wiles [214].
3.6.2. Weil’s converse theorem. We can read Hecke’s theorem 3.9 as
follows: suppose f (z) is given by a Fourier series with polynomially bounded
coefficients a(n). Then f (z) is a modular form of weight k for the full modular
group if and only if the function
∞
X
a(n)
−s
Φ(s) = (2π) Γ(s)
ns
n=1
has an analytic continuation to C such that
Φ(s) +
a(0) ik a(0)
+
s
s−k
Chapter 3
138
Hecke theory
is entire and bounded in any vertical strip and satisfies the functional equation
Φ(s) = ik Φ(k − s).
However, if we are dealing with modular forms to congruence subgroups, then
more than one functional equation is needed in order to show modularity.
This follows from the fact that in general there are more pairs of modular
forms and associated Dirichlet series if the level q of the congruence subgroup
Γ0 (q) is large, and thus we cannot easily identify one single pair. We observe
that, fortunately, there are many characters χ mod q which can be used to
find additional functional equations. This idea is due to Weil [209] who
proved (a stronger version of)
Theorem 3.10. Let k, N ∈ N and ǫ ∈ {±1}. Given a sequence of complex
numbers a(n) ≪ nc , define
f (z) =
∞
X
a(n) exp(2πinz).
n=1
Assume that the function
−s
Φ(s) := (2π) Γ(s)L(s),
where L(s) :=
∞
X
a(n)
n=1
ns
,
has the property that
a(0) ǫa(0)
+
s
k−s
defines an entire function which is bounded in any vertical strip and satisfies
the functional equation
Φ(s) +
Φ(s) = ǫΦ(k − s).
Finally, suppose that for any primitive character χ mod m and m coprime
with N, the function
√ !s
∞
X
a(n)χ(n)
m N
,
Γ(s)Lχ (s),
where Lχ (s) :=
Φχ (s) :=
s
2π
n
n=1
extends to an entire function which is bounded in any vertical strip and satisfies the functional equation
Φχ (s) = ǫχ(−N)
τ (χ)
Φχ (k − s),
τ (chi)
where τ (χ) is the Gauss sum associated to χ. Then, f is a modular form of
weight k for Γ0 (N). If in addition the series defining L(s) converges absolutely for σ > k − ε for some ε > 0, then f is a cusp form.
Section 3.6
Shimura-Taniyama-Wiles
139
The proof is beyond the scope of our notes (and uses much of the theory of
Hecke operators); it can be found in Iwaniec [99] and Ogg [163].
Weil’s converse theorem gave support to the Shimura-Taniyama conjecture
(and in some literature this conjecture is also named Shimura-Taniyama-Weil
conjecture). The conjecture of Hasse stated that the L-function LE (s) of an
elliptic curve satisfies a functional equation of the form as the one for Dirichlet
series attached to modular forms of weight k = 2. Furthermore, L-functions
to newforms have an Euler product comparable to the one of L-functions to
elliptic curves. The famous Shimura-Taniyama conjecture claims that indeed
these two objects from different fields are just the same: for any elliptic curve
E, there exists a newform f of weight 2 for some congruence subgroup Γ0 (N)
such that LE (s) = L(s, f ). In many instances one can use Weil’s converse
theorem to verify that LE (s) is indeed the L-function to such a newform. In
the next section we shall briefly discuss one example.
At the end of his paper [209] Weil restates the Shimura-Taniyama conjecture with respect to some correspondence from Shimura (Taniyama committed suicide in 1957) and writes
Ob die Dinge immer (. . .) sich so verhalten, scheint im Moment
noch problematisch zu sein und mag dem interessierten Leser
als Übungsaufgabe empfohlen werden.
3.6.3. An example. We shall illustrate Weil’s converse theorem with
an example (following Iwaniec [99] and Koblitz [117]). For a square-free
integer m, consider the family of elliptic curves Em given by the equations
Em :
Y 2 = X 3 − m2 X.
The discriminant is easily seen to be ∆m = (2m)6 . For any prime p we obtain
by reduction modulo p reduced curves Em (Z/pZ) and the number of points
on these elliptic curves is given by λm (p) = p + 1 − ♯Em (Z/pZ). For fixed p
we consider the congruence
Y 2 ≡ X 3 − m2 X mod p.
It is easily seen that there are no solutions for p = 2. For any odd prime we
count the solutions in terms of the Legendre symbol. This leads to
X 3
X x3 − m2 x x −x
m
=−
,
λm (p) = −
p
p x mod p
p
x mod p
where the identity in the last step comes from the substitution x 7→ mx.
Hence,
m
λm (p) =
λ1 (p).
p
Chapter 3
140
Hecke theory
Following Hasse’s construction, we find
−1
Y
λ1 (p) m
×
1− s
LEm (s) =
p
p
p|∆m
×
Y
p∤∆m
λ1 (p)
1− s
p
m
p
+
1
p2s−1
m
p
2 !−1
.
In the half-plane σ > 23 we may expand the Euler product into a Dirichlet
series and it follows that
∞
∞
X
λm (n) X λ1 (n) m ;
=
LEm (s) =
s
s
n
n
n
n=1
n=1
here the values λm (n) are defined by multiplicativity. The right-hand side is
the Dirichlet series of LE1 (s) twisted by the Jacobi symbol:
ν
m
Y
m
=
,
n 7→
n
pj
j=1
p1 ·...·pν =n
where the pj are the (not necessarily distinct) prime divisors of n. We notice
this fact as
m LEm (s) = LE1 s,
.
The Jacobi symbol m is the quadratic character corresponding to the num√
ber field Q( m); in particular, it is a primitive character.
We consider now the special case m = 1. It is not difficult to see that
λ1 (p) = 0 for p ≡ 3 mod 4. One can show that the primes p ≡ 1 mod 4 split
√
in Z[ −1] into p = ππ and
λ1 (p) = −(π + π).
(3.44)
For this aim one has to consider the bijection
φ : E1 \ (0, 0)
→
E′ : Y 2 = X 4 + 1
(x, y) 7→ (yx−1 , 2x − y 2x−2 ).
it follows that
♯E1 (Z/pZ) = ♯E ′ (Z/pZ) − 1.
Since (Z/pZ)∗ is cyclic and dual to the character group mod p,
X
♯{x ∈ (Z/pZ)∗ : x4 = z} =
χ(z)
χ4 =1
for any z ∈ (Z/pZ)∗ . Hence,
X
G(χ)
♯E ′ (Z/pZ) =
χ4 =1
with G(χ) :=
X
y mod p
χ(y 2 − 4).
Section 3.6
Shimura-Taniyama-Wiles
141
One can show that there are four characters with χ4 = 1, namely, the powers
1, ψ, ψ 2, and ψ 4 of the Legendre symbol ψ(n) = ( np ). It is easily computed
that G(1) = p − 2, G(ψ 2 ) = −1, G(ψ 3 ) = G(ψ), and
X y2 − 4 G(ψ) =
= π.
p
y mod p
Hence,
λ1 (p) = p + 2 − G(1) + G(ψ) + G(ψ 2 ) + G(ψ 3 ) = −(π + π);
this is (3.44). Thus, we find
Y −1
Y π+π
p
p
1+
1 + 2s
+ 2s
,
LE1 (s) =
p
ps
p
p≡1 mod 4
p≡3 mod 4
p=ππ
resp.
−1
Y
χ(p)
LE1 (s) =
1−
N(p)−s
p∤2
√
for some grössencharacter χ on Q( −1), where the product is taken over
the prime ideals p coprime with 2. Grössencharacters are a generalization
of Dirichlet characters and the L-function above is an example of a so-called
Hecke L-function associated with number fields, being an analogue of Dirichlet L-functions. These L-functions are known to have analytic continuation
and a functional equation (and we shall study them more detailed in the
next chapter). An application of Weil’s converse theorem 3.10 now yields
that LE1 (s) = L(s, f ), where
(3.45)
∞
X
f (z) =
λ1 (n)q n = q − 2q 5 − 3q 9 + 6q 13 + . . .
with q = exp(2πiz)
n=1
defines a newform of weight 2 for the congruence subgroup Γ0 (32). Moreover,
one can show that LEm (s) = L(s, fm ) with
fm ∈ S2 (Γ0 (32m2 )) or
∈ S2 (Γ0 (16m2 ))
according to m odd or even. These observations date back at least to Tunnell [201] and his study on the congruent number problem. More generally,
one can show that a cusp form of weight k, level N and multiplier χ of conductor q, twisted with a character ψ mod r, is a cusp form of weight k, level
lcm[N, qr, r 2 ] (the least common multiple) and multiplier χψ 2 . Twisting with
quadratic characters is a well-known technique in the theory of elliptic curves,
and the example of the elliptic curves Em is only one of many.
Before 1995, the Shimura-Taniyama conjecture was known for elliptic
curves with complex multiplication and in isolated examples. However, by
142
Chapter 3
Hecke theory
the work of Wiles & Taylor [213, 198] and Breuil et al. [26] it is now a
theorem:
Modularity theorem. For any elliptic curve E, there exists a newform f
of weight 2 for some congruence subgroup Γ0 (N) such that LE (s) = L(s, f ).
Exercise 88. * Let E be an elliptic curve and p prime. Consider the sum
X x3 + ax + b λ(p) := p + 1 − ♯E(Z/pZ) =
p
x mod p
as a random walk. What would the theory of random walks imply?
Exercise 89. Let E be an elliptic curve. Prove that the Euler product LE (s)
converges for σ > 23 .
Exercise 90. ** Read in Iwaniec [99] and Koblitz [117] and fill the gaps in the
proof of LE1 (s) = L(s, f ), where f is given by (3.45).
CHAPTER 4
The Selberg class – an axiomatic approach
“What is an L-function? We know it when we see one!”
M.N. Huxley.
In view of plenty of examples of Dirichlet series in arithmetic it might be
reasonable to ask for a classification and to search for common patterns in
their analytic properties. There were several noticeable attempts to define
classes of relevant Dirichlet series (as for example Lekkerkerker [129], Perelli
[166], and Matsumoto [143]), however, these classes were in some sense lacking algebraic structure.
In 1989, Selberg [180] defined a general class of Dirichlet series having an
Euler product, analytic continuation and a functional equation of Riemanntype (plus some side conditions), and formulated some fundamental conjectures concerning them. Especially these conjectures give this class of Dirichlet
series a certain structure which applies to central problems in number theory.
He writes about his conjectures that
“these conjectures, which, by the way, are not unrelated to several other conjectures like the Sato-Tate conjecture, Langlands
conjectures, etc., have been verified in a number of cases for
Dirichlet series with functional equation and Euler product that
occur in number theory, by assuming that the factorizations we
can give are actually that a function is really primitive and
cannot be factorized further.”
Indeed, one of its consequences is the famous yet unsolved Artin conjecture.
In the meantime this so-called Selberg class became an important object
of research but still it is not understood very well. It is conjectured that the
Selberg class consists of the automorphic L-functions and that the analogue
of the Riemann hypothesis holds for all its elements.
4.1. Definition and first observations
The Selberg class S consists of Dirichlet series
L(s) :=
satisfying the following hypotheses:
∞
X
a(n)
n=1
143
ns
Chapter 4
144
The Selberg class
• Ramanujan hypothesis: a(n) ≪ nǫ for any ǫ > 0;
• Analytic continuation: there exists a non-negative integer k such
that (s − 1)k L(s) is an entire function of finite order;
• Functional equation: there exists a positive integer f , and for
1 ≤ j ≤ f , there are positive real numbers Q, λj , and there are
complex numbers µj , ω with Re µj ≥ 0 and |ω| = 1, such that
ΛL (s) = ωΛL (1 − s),
where
ΛL (s) := L(s)Qs
f
Y
Γ(λj s + µj );
j=1
• Euler product: L(s) satisfies
Y
L(s) =
Lp (s),
p
where
Lp (s) = exp
∞
X
b(pk )
k=1
pks
!
with suitable coefficients b(pk ) satisfying b(pk ) ≪ pkθ for some θ < 21 .
The Ramanujan hypothesis implies that the Dirichlet series converges absolutely in the half-plane σ > 1, and uniformly in every compact subset. Thus
it follows that elements L(s) are analytic in σ > 1 and so it makes sense to
speak about analytic continuation. The axiom on the Euler product implies
that the coefficients a(n) are multiplicative, and that each Euler factor has
the Dirichlet series representation
Lp (s) =
∞
X
a(pk )
k=0
pks
,
absolutely convergent for σ > 0.
Obvious examples in the Selberg class are the Riemann zeta-function ζ(s)
and Dirichlet L-functions L(s, χ) to primitive characters; notice that L(s, χ)
with a non-primitve character χ mod q, q 6= 1, is not in S by lack of the correct form of the functional equation. More advanced examples are Dedekind
zeta-functions. Kaczorowski et al. [105] studied Hecke L-functions to modular forms of Hecke groups; they were shown to be either in the Selberg class
or a related class where the axiom of the functional equation is adjusted.
In view of the Euler product representation it is clear that any element
L(s) of the Selberg class does not vanish in the half-plane of absolute convergence σ > 1. This gives rise to the notions of critical strip and critical
line. The zeros of L(s) located at the poles of gamma-factors appearing in
Section 4.1
Definition
145
the functional equation are called trivial. They all lie in σ ≤ 0, and it is
easily seen that they are located at
(4.1)
s=−
k + µj
λj
with k ∈ N0
and 1 ≤ j ≤ f.
All other zeros are said to be nontrivial and they lie in the critical strip
0 ≤ σ ≤ 1. In general we cannot exclude the possibility that L(s) has a
trivial zero and a nontrivial one at the same point. It is expected that for
every function in the Selberg class the analogue of the Riemann hypothesis
holds, that is all nontrivial zeros lie on the critical line:
Grand Riemann hypothesis. If L ∈ S, then L(s) 6= 0 for σ > 12 .
Following Conrey & Ghosh [38] we motivate the axioms defining S. We have
already seen that the Ramanujan hypothesis implies the regularity of L(s)
in σ > 1. Further we note:
• The condition that there be at most one pole, and that this one is
located at s = 1, is natural. If we would allow more poles they would
lie on the line σ = 1, and for each of them L(s) we would expect
the zeta-function suitably shifted as a factor (since otherwise L(s)
would have zeros off the critical line). It is now obvious that it is
sufficient to investigate functions with at most one pole, normalized
to be at s = 1.
• The restriction Re µj ≥ 0 in the functional equation comes from
the theory of Maass waveforms. Assume that there exists an arithmetic subgroup of SL2 (R) together with a Maass cusp form that
corresponds to an exceptional eigenvalue, and suppose that the
Ramanujan-Petersson conjecture holds, then the L-function associated with the Maass cusp form has a functional equation with µj
which satisfies Re µj < 0, but the L-function violates Riemann’s
hypothesis.
• Finally, consider the axiom concerning the Euler product. It is wellknown that the existence of an Euler product is a necessary (but not
sufficient) condition for Riemann’s hypothesis. On the first sight the
condition θ < 12 seems to be a little bit unnatural. However, if θ = 21
would be allowed, the function
(1 − 2
1−s
)ζ(s) =
∞
X
(−1)n−1
n=1
ns
would lie in S, but obviously, it violates Riemann’s hypothesis (see
also the proof of Theorem 4.1 below; further examples were given by
Kaczorowski & Perelli [106]).
146
Chapter 4
The Selberg class
The zero-distribution is essential for the Selberg class. If anyone of the discussed restrictions would be removed, the resulting larger class would probably contain Dirichlet series for which the Riemann hypothesis does not hold.
Exercise 91. Assume that L ∈ S and let θ be a fixed real number. Show that if
L(s) is regular at s = 1, then also L(s + iθ) is an element of S.
Exercise 92. Verify that Dedekind zeta-functions are elements of the Selberg class.
4.2. The structure of the Selberg class
The structure of the Selberg class is of special interest. Obviously, the
Selberg class is multiplicatively closed. To classify its finer structure we need
a quantity in order to measure the size of its elements.
The degree of L ∈ S is defined by
dL = 2
f
X
λj ,
j=1
where the λj are from the Gamma-factors in the functional equation. Although the data of the functional equation is not unique, the quantity dL is
well-defined. If NL (T ) counts the number of zeros of L ∈ S in the rectangle
0 ≤ σ ≤ 1, |t| ≤ T (according to multiplicities) one can show by standard
contour integration
dL
T log T
π
in analogy to the Riemann-von Mangoldt formula (1.12) for Riemann’s zetafunction; we shall give a more precise asymptotic formula in Theorem 4.11
below. It is conjectured that all L ∈ S have integral degree. This is the
degree conjecture. Slightly stronger is the
(4.2)
NL (T ) ∼
Strong λ-conjecture. Let L ∈ S. All λj appearing in the gamma-factors
of the functional equation can be chosen to be equal to 21 .
4.2.1. The case of small degrees. Recently, Kaczorowski & Perelli
[109] showed that all functions L ∈ S with degree 0 < dL < 53 have degree
equal to one. This supports the degree conjecture; moreover, they obtained
a complete classification of all elements in the Selberg class of degree d < 53
and for all of them it turned out that also the strong λ-conjecture is true.
Here we shall only prove
Section 4.2
Structure of the Selberg class
147
Theorem 4.1. Let L ∈ S. If dL = 0, then L(s) ≡ 1. If dL is positive, then
dL ≥ 1.
This weaker statement was first proved by Conrey & Ghosh [38]; however, it
is essentially included in Bochner’s extension of Hamburger’s theorem 3.8 on
the Riemann functional equation [18] (see also Vignéras [203]). For the first
statement we follow the argument of Conrey & Ghosh, and for the second
claim we follow Molteni [147].
Proof. We may assume that dL < 1. Let B be a constant such that
a(n) ≪ nB . By Perron’s formula (1.36), we find
c+B Z c+iT
X
xs
x
1
,
L(s) ds + O
a(n) =
2πi
s
T
c−iT
n≤x
where c > 1 is a constant. Shifting the path of integration to the left, yields,
by the Phragmén-Lindelöf principle (see Section 3.1.7 and the following section), the asymptotic formula
X
d −1
(1+B) dL +1 +ǫ
L
a(n) = xP (log x) + O x
,
n≤x
where P (x) is a computable polynomial according to the residue of L(s) at
s = 1. By subtraction, this implies
(4.3)
(1+B)
a(n) ≪ n
dL −1
+ǫ
dL +1
,
where the implicit constant depends on B. For dL < 1 the exponent is
negative, and we may choose B arbitrarily large. Then L(s) is uniformly
bounded in every right half-plane. This is a contradiction for L ∈ S with
positive degree since the functional equation implies a certain order of growth
(this will become clear in the following section). This shows that S is free of
elements having degree 0 < d < 1.
It remains to consider the case that dL = 0. Then the functional equation
takes the form:
Qs L(s) = ωQ1−s L(1 − s)
(there are no Gamma-factors). By (4.3) the a(n) are so small that the Dirichlet series for L(s) converges in the whole complex plane. Thus we may rewrite
the functional equation as
2 s
∞
∞
X
X
a(n) s
Q
= ωQ
n.
(4.4)
a(n)
n
n
n=1
n=1
We may regard this as an identity between absolutely convergent Dirichlet
series. Thus, is a(n) 6= 0, then Q2 /n is an integer. In particular, q := Q2 ∈ N.
Moreover, since Q2 has only finitely many divisors, it follows that L(s) is a
Chapter 4
148
The Selberg class
Dirichlet polynomial. If q = 1, then L(s) ≡ 1 and we are done with the case
dL = 0. Hence, we may assume that q > 1.
Since the Dirichlet coefficients a(n) are multiplicative, we have a(1) = 1
and via (4.4)
a(1)Q2s = ωQ−1 a(Q2 )Q2s ;
thus, |a(q)| = Q. In particular, there exists a prime p such that the exponent
ν of p in the prime factorization of q is positive and, by the multiplicativity
of the a(n)’s,
ν
|a(pν )| ≥ p 2 .
Now consider the logarithm of the corresponding Euler factor:
!
ν
∞
m
X
X
a(p )
b(pk )
log 1 +
=
.
pms
pks
m=1
k=1
Viewing this as power series in X = p−s , we write
∞
X
log P (X) =
Bk X k
with Bk = b(pk ).
k=1
Since a(1) = 1 we find
P (X) = 1 +
ν
X
m
a(p )X
m
=
m=1
Now
ν
Y
j=1
ν
Y
j=1
ν
1X k
C .
with Bk = −
k j=1 j
(1 − Cj X)
ν
|Cj | = |a(pν )| ≥ p 2 ,
1
and thus the maximum of the values |Cj | is greater than or equal to p 2 . We
have
1
ν
k
1 X
1
Cjk = max |Cj |;
lim |b(pk )| k = lim 1≤j≤ν
k→∞
k→∞ k
j=1
by our foregoing observations the right-hand side is greater than or equal to
1
p 2 . This is a contradiction to the condition b(pk ) ≪ pkθ with some θ < 12 in
the axiom on the Euler product. Hence, q = 1 and L(s) ≡ 1. This proves
the first statement.
The theorem is proved. •
By the work of Kaczorowski & Perelli [107] it is known that the functions
of degree one in the Selberg class are the Riemann zeta-function and shifts
L(s + iθ, χ) of Dirichlet L-functions attached to primitive characters χ with
θ ∈ R. However, for higher degree there is no complete classification so far.
Examples of degree two are normalized L-functions associated with holomorphic newforms; here the notion normalized means that a(p) is replaced by
Section 4.3
The Riemann–von Mangoldt formula
149
k−1
a(p)p− 2 in the notation of Section 3.4. Normalized L-functions attached
to non-holomorphic newforms are expected to lie in S but the Ramanujan
hypothesis is not yet verified. The Rankin-Selberg L-function of any two
holomorphic newforms is an element of the Selberg class of degree 4. Other
examples are Dedekind zeta-functions to number fields K; their degree is
equal to the degree of the field extension K/Q.
Exercise 93. Prove that the Selberg class is multiplicatively closed.
Exercise 94. Show that the data of the functional equation is not unique.
Hint: Legendre’s duplication formula for the Gamma-function.
Exercise 95. Verify that L-functions associated with newforms are elements of
the Selberg class.
4.3. The Riemann–von Mangoldt formula
Riemann conjectured an asymptotic formula for the number N(T ) of nontrivial zeros ρ = β + iγ of ζ(s) with 0 ≤ γ ≤ T (counted according multiplicities). This so-called Riemann-von Mangoldt formula (1.12) was proved
by von Mangoldt in 1895. Now we want to show a Riemann-von Mangoldt
formula for elements of the Selberg class. One method is contour integration applied to the logarithmic derivative. This is the classic approach due
to von Mangoldt and it can be found in Titchmarsh [200] and many other
books for the special case of the zeta-function. Here we shall go another way
which provides more information on the value-distribution of the L-functions
in question. This method is due to Levinson [133] who applied it to the
zeta-function; the application to the Selberg class is from Steuding [192].
However, first of all we have to state some preliminary results (not all with
proofs).
4.3.1. Mean-square estimates. The order of growth of a meromorphic
function is of special interest. Recall our observations on the order of growth
of Dirichlet series from Section 3.1.7. For L ∈ S we define
log |L(σ + it)|
.
µL (σ) = lim sup
log |t|
t→±∞
One can show that µL (σ) is a convex function of σ. Taking into account the
absolute convergence of the defining Dirichlet series we obtain immediately
µL (σ) = 0 for σ > 1. The order of growth in the half-plane left of the critical
strip is ruled by the functional equation which we may rewrite as
(4.5)
L(s) = ∆L (s)L(1 − s),
150
Chapter 4
The Selberg class
where
1−2s
∆L (s) := ωQ
f
Y
Γ(λj (1 − s) + µj )
j=1
Γ(λj s + µj )
.
Applying Stirling’s formula (1.53), we get after a short computation
Lemma 4.2. Let L ∈ S. For t ≥ 1, uniformly in σ,
1
1
iπ(µ − dL )
2 dL 2 −σ−it
ω+O
,
exp it dL +
∆L (σ + it) = λQ t
4
t
where
f
f
Y
X
2λ
λj j .
(1 − 2µj )
and
λ :=
µ := 2
j=1
j=1
Using the Phragmén-Lindelöf principle, we can obtain upper bounds for the
order of growth inside the critical strip.
Theorem 4.3. Let L ∈ S. Uniformly in σ, as |t| → ∞,
1
L(σ + it) ≍ |t|( 2 −σ) dL |L(1 − σ + it)|.
In particular,


0
µL (σ) ≤
d (1 − σ)
 1L
( 2 − σ) dL
1
2
if
σ > 1,
if 0 ≤ σ ≤ 1,
if
σ < 0.
This theorem should be compared with our results for the zeta-function from
the previous chapters. Our proof is more or less the same as in Section 3.1.7.
Proof. The first assertion follows immediately from the functional equation
and Lemma 4.2. This estimate implies for σ < 0
1
− σ dL .
µL (σ) =
2
The calculation of µL (σ) for 0 ≤ σ ≤ 1 is more difficult. Here we apply the
theorem of Phragmén-Lindelöf, Lemma 3.5. In view of the axiom concerning
the analytic continuation L(s) is a function of finite order. Thus, Lemma 3.5
shows that µL (σ) is non-increasing and convex downwards. By the estimates
of µL (σ) for σ outside of the critical strip the second assertion of the theorem
follows. •
It should be noticed that we did not use the condition that the µj appearing
in the gamma factors of the functional equation have positive real part.
In view of the functional equation, resp. the convexity of µL , the value for
σ = 21 is essential. In particular, we obtain µL ( 21 ) ≤ 14 dL , or equivalently,
1
1
L
+ it ≪ |t| 4 dL +ǫ ,
(4.6)
2
Section 4.3
The Riemann–von Mangoldt formula
151
valid for |t| ≥ 1.
Next we shall apply the following general theorem on the mean-square of
Dirichlet series satisfying a Riemann-type functional equation due to Potter
[171].
Theorem 4.4. Suppose that the functions
∞
∞
X
X
bn
an
and
B(s) =
A(s) =
s
n
ns
n=1
n=1
have a half-plane of convergence, are of finite order, and that all singularities lie in a subset of the complex plane of finite area. Further, assume the
estimates
X
X
|an |2 ≪ xb+ǫ
and
|bn |2 ≪ xb+ǫ ,
n≤x
n≤x
as x → ∞, and that A(s) and B(s) satisfy
A(s) = h(s)B(1 − s),
c( a2 −σ)
uniformly in σ for σ from a finite interval, as |t| → ∞,
where h(s) ≍ |t|
and c is some positive constant. Then
Z T
∞
X
1
|an |2
2
lim
|A(σ + it)| dt =
T →∞ 2T −T
n2σ
n=1
for σ > max{ a2 , 12 (b + 1) − 1c }.
We do not give the lengthy proof of Potter’s theorem here and refer directly
to Potter [171]. But we shall apply the theorem to L-functions in the Selberg
class. Taking into account Lemma 4.3 we obtain
n
o
Corollary 4.5. Let L ∈ S. For σ > max 21 , 1 − d1L ,
1
lim
T →∞ 2T
Z
T
−T
2
|L(σ + it)| dt =
∞
X
|a(n)|2
n=1
n2σ
.
Note that the series on the right hand side converges on behalf of the Ramanujan hypothesis (resp. the polynomial Euler product representation).
Every (convergent) Dirichlet series has a mean-square half-plane (see Titchmarsh [199]), i.e., a half-plane in which the mean-square on vertical lines is
bounded. In view of Corollary 4.5 the mean-square half-plane of L ∈ S
contains the region
1
1
.
,1 −
σ > max
2
dL
It is expected that the mean-square exists for any L ∈ S for σ > 12 (as in the
case of zeta). However, this is a deep conjecture and its verification is even
in single cases a difficult task. In fact the difficulties arise for large degrees
152
Chapter 4
The Selberg class
dL . Potter’s theorem yields only an asymptotic formula throughout σ > 12
if the degree dL is less than or equal to two. The difficulties become more
obvious by noting that any result on the mean-square of an L-function from
the Selberg class of degree d is comparable to the corresponding result for
the 2 d-th moment of the Riemann zeta-function.
4.3.2. Sums over c-values. Let c be a complex number. Levinson [133]
(T )
of the roots of ζ(s) = c in T < t < 2T lie in
proved that all but ≪ logN log
T
2
1
σ − < (log log T ) .
2
log T
Thus, the c-values of the zeta-function are clustered around the critical line.
In particular, we see that the density estimate 2.16 does not indicate the
truth of the Riemann hypothesis. As we shall show now, this distribution of
c-values is typical for L-functions in the Selberg class.
The c-values of L(s) are the roots of the equation
L(s) = c,
(4.7)
which we denote by ρc = βc + iγc . Our first aim is to prove estimates for
sums taken over c-values, weighted with respect to their real parts.
Theorem 4.6. Let L ∈ S˜ and c 6= 1. Then, for b > max{ 12 , 1 −
X
(βc − b) ≪ T.
1
},
dL
βc >b
T <γc ≤2T
Assuming the truth of Lindelöf ’s hypothesis, i.e.,
1
L
+ it ≪ tǫ
2
as t → ∞, we have
X 1
βc > 2
T <γc ≤2T
1
βc −
2
= o(T log T ).
The case c = 1 is exceptional since 1 is the limit of L(s) as σ → ∞:
(4.8)
L(s) = 1 + O(2−σ ).
However, without big effort one can obtain also in this case similar estimates.
It should be noted that the Lindelöf hypothesis for L(s) follows from the
Riemann hypothesis and thus it is widely expected to hold.
Proof of Theorem 4.6. In view of (4.8) there exists a positive real number
A depending on c such that all real parts βc of c-values satisfy βc < A. Put
L(s) − c
.
ℓ(s) =
1−c
Section 4.3
The Riemann–von Mangoldt formula
153
Obviously, the zeros of ℓ(s) correspond exactly to the c-values of L(s). Next
we apply Littlewood’s lemma 2.13. Let ν(σ, T ) denote the number of zeros
ρc of ℓ(s) with βc > σ and T < γc ≤ 2T (counting multiplicities). Let a be a
parameter with a > max{A + 1, b}. Then Littlewood’s lemma 2.13, applied
to the rectangle R with vertices a + iT, a + 2iT, b + iT, b + 2iT , gives
Z
Z a
log ℓ(s) ds = −2πi
ν(σ, T ) dσ.
R
b
Since
(4.9)
Z
a
X Z
ν(σ, T ) dσ =
b
βc >b
T <γ≤2T
b
βc
dσ =
X
βc >b
T <γc ≤2T
(βc − b),
we get
2π
X
βc >b
T <γc ≤2T
Z
(βc − b) =
log |ℓ(b + it)| dt −
T
−
(4.10)
2T
Z
4
X
=
Z
a
arg ℓ(σ + iT ) dσ +
b
2T
log |ℓ(a + it)| dt +
T
Z
a
arg ℓ(σ + 2iT ) dσ
b
Ij ,
j=1
say. To define log ℓ(s) we choose the principal branch of the logarithm on the
real axis, as σ → ∞; for other points s the value of the logarithm is obtained
by analytic continuation.
We start with the vertical integrals. Obviously,
Z 2T
(4.11) I1 (T, b) := I1 =
log |L(b + it) − c| dt − T log |1 − c|.
T
By Jensen’s inequality the integral is
Z 2T
1
T
2
|L(b + it)| dt + O(T ).
≤ log
2
T T
By Corollary 4.5 this is ≪ T for b > max{ 21 , 1 − d1L }. Thus we get I1 (T, b) ≪
T unconditionally. An immediate consequence of Lindelöf’s hypothesis is
2
Z 2T 1
L
+ it dt ≪ T 1+ǫ
2
T
for any positive ǫ. Thus, assuming the truth of Lindelöf’s hypothesis we get
1
≪ ǫT log T.
I1 T,
2
Chapter 4
154
The Selberg class
Next we consider I2 . Since a > 1 we have
∞
1 X a(n)
,
ℓ(a + it) = 1 +
1 − c n=2 na+it
(4.12)
and in view of (4.8) the absolute value of the series is less than 1 for sufficiently
large a. Therefore we find by the Taylor expansion of the logarithm
log |ℓ(a + it)| = Re
∞
∞
∞
X
X
a(n1 ) · . . . · a(nk )
(−1)k X
.
.
.
.
k
a+it
k(1
−
c)
(n
·
.
.
.
·
n
)
1
k
n =2
n =2
k=1
1
k
This leads by the Ramanujan hypothesis to the estimate
I2
(4.13)
∞
∞
∞
X
X
(−1)k X
a(n1 ) · . . . · a(nk )
.
.
.
×
k
a
k(1
−
c)
(n
·
.
.
.
·
n
)
1
k
n1 =2
nk =2
k=1
Z 2T
dt
×
(n1 · · · nk )it
T
!k
∞
∞
X
1 X 1
≪ 1,
≪
a−ǫ
k
n
n=2
k=1
= Re
for sufficiently large a. It remains to estimate the horizontal integrals I3 , I4 .
Suppose that Re ℓ(σ + iT ) has N zeros for b ≤ σ ≤ a. Then divide [b, a]
into at most N + 1 subintervals in each of which Re ℓ(σ + iT ) is of constant
sign. Then
| arg ℓ(σ + iT )| ≤ (N + 1)π.
(4.14)
To estimate N let
1
g(z) =
ℓ(z + iT ) + ℓ(z + iT ) .
2
Then we have g(σ) = Re ℓ(σ + iT ). Let R = a − b and choose T so large that
T > 2R. Now, Im (z + iT ) > 0 for |z − a| < T . Thus ℓ(z + iT ), and hence
g(z) is analytic for |z − a| < T . Let n(r) denote the number of zeros of g(z)
in |z − a| ≤ r. Obviously, we have
Z 2R
Z 2R
n(r)
dr
dr ≥ n(R)
= n(R) log 2.
r
r
0
R
With Jensen’s formula, Lemma 1.17,
Z 2π
Z 2R
1
n(r)
log g a + 2Reiθ dθ − log |g(a)|,
(4.15)
dr =
r
2π 0
0
we deduce
1
n(R) ≤
2π log 2
Z
2π
0
log |g(a)|
log g a + 2Reiθ dθ −
.
log 2
Section 4.3
The Riemann–von Mangoldt formula
155
By (4.12) it follows that log |g(a)| is bounded. By Theorem 4.3, in any vertical
strip of bounded width,
L(s) ≪ |t|B
with a certain positive constant B. Obviously, the same estimate holds for
g(z). Thus, the integral above is ≪ log T , and n(R) ≪ log T . Since the
interval (b, a) is contained in the disc |z − a| ≤ R, the number N is less or
equal n(R). Therefore, with (4.14), we get
Z a
|I4 | ≤
| arg ℓ(σ + iT )| dσ ≪ log T.
b
Obviously, I3 can be bounded in the same way.
Collecting all estimates, the assertions of the theorem follow. •
Now we will include most of the c-values into our observations. In view
of Theorem 4.3 there exist positive constants C ′ , T ′ such that there are no
c-values in the region σ < −C ′ , t ≥ T ′ . Therefore, assume that b < −C ′ − 1
and T ≥ T ′ + 1. By the functional equation in the form (4.5),
!
1
.
log |L(s) − c| = log |∆L (s)| + log |L(1 − s)| + O
|∆L (s)L(1 − s)|
In view of Lemma 4.2
log |∆L (s)| =
1
1
2
.
− σ ( dL log t + log(λQ )) + O
2
t
Thus
Z
2T
log |L(b + it) − c| dt
Z 2T
1
=
−b
( dL log t + log(λQ2 )) dt
2
T
Z 2T
+
log |L(1 − b − it)| dt + O(log T ).
T
T
Now suppose that c 6= 1. The first integral on the right hand side is easily
calculated by elementary means. The second integral is small if −b is chosen
sufficiently large (see (4.13)). Thus, taking into account (4.10) and (4.11),
we get
4T
1
2
−b
dL T log
+ T log(λQ ) − T log |1 − c| + O(log T ).
I1 =
2
e
By (4.10) and with the estimates for the Ij ’s from the proof of the previous
theorem we obtain
Chapter 4
156
The Selberg class
Theorem 4.7. Let L ∈ S̃ and c 6= 1. Then, for sufficiently large negative b,
X
1
4T
2
2π
(βc − b) =
−b
dL T log
+ T log(λQ )
2
e
T <γ ≤2T
c
−T log |1 − c| + O(log T ).
4.3.3. Riemann-von Mangoldt-type formulae.
sum over c-values from the previous section as follows:
X
X
X
1
βc −
1+
−b
(βc − b) =
2
β
β
β
c
c
c
We can rewrite the
1
.
2
The first sum on the right counts the number of c-values and the second one
measures the distances of the c-values from the critical line. Let N c (T ) count
the number of c-values of L(s) with T < γc ≤ 2T . Then, subtracting the
formula of Theorem 4.7 with b + 1 instead of b from the one with b, we obtain
Corollary 4.8. Let L ∈ S̃. Then, for c 6= 1,
N c (T ) =
4T
T
dL
T log
+
log(λQ2 ) + O(log T ).
2π
e
2π
Furthermore,
Corollary 4.9. Let L ∈ S̃. Then, for c 6= 1,
X 1
T
βc −
= − log |1 − c| + O(log T ).
2
2π
T <γ ≤2T
c
Thus, for c satisfying |1 − c| =
6 1, the c-values, weighted with respect to
their distance to the critical line, lie asymetrically distributed (which is not
too surprising in view of the fact that µL (σ) is increasing as σ → −∞).
Nevertheless, our next aim is to show that most of the c-values lie close
to the critical line. Unfortunately, for this purpose we have to assume the
Lindelöf hypothesis. Define the counting functions (according multiplicities)
N+c (σ, T ) = ♯{ρc : T < γc ≤ 2T, βc > σ},
and
Then
N−c (σ, T ) = ♯{ρc : T < γc ≤ 2T, βc < σ}.
Theorem 4.10. Let L ∈ S˜ and c 6= 1. Then, for any σ > max{ 21 , 1 −
(4.16)
N+c (σ, T ) ≪ T,
and assuming the truth of the Lindelöf hypothesis, for any δ > 0,
1
1
c
c
N−
− δ, T + N+
+ δ, T ≪ δT log T.
2
2
1
},
dL
Section 4.3
The Riemann–von Mangoldt formula
157
Proof. First of all, let σ > max{ 12 , 1 − d1L } and fix σ1 ∈ (max{ 21 , 1 −
Then
X
1
N+c (σ, T ) ≤
(βc − σ1 ).
σ − σ1 βc >σ
1
}, σ).
dL
T <γc ≤2T
The sum on the right side is less than or equal to
Z 2T
X
(βc − σ1 ) ≪
log |ℓ(σ1 + it)| dt + O(log T ),
T
βc >σ1
T <γc ≤2T
where we used Littlewood’s lemma 2.13 and the techniques from the previous
section for the latter inequality. In view of the unconditional estimate for
(4.11) in the proof of Theorem 4.6 we obtain (4.16). Assuming the truth of
the Lindelöf hypothesis we get analogously
1
ǫ
c
N+
+ δ, T ≪ T log T
2
δ
for any positive ǫ.
Next we consider N−c ; in particular, we assume the Lindelöf hypothesis for
L(s). Let b be a sufficiently large constant. We have
X
X
X 1
1
(βc − b) ≤
1+
βc −
.
−b
2
2
1
1
1
βc ≥ 2 −δ
T <γc ≤2T
βc ≥ 2
T <γc ≤2T
βc ≥ 2 −δ
T <γc ≤2T
Hence
X
T <γc ≤2T
(βc − b) =
≤
X 1
1
+
− b + βc −
2
2
1
βc < 2 −δ
T <γc ≤2T
+
1
− b N c (T ) +
2
X 1
βc > 2
T <γc ≤2T
1
βc −
2
X
βc ≥ 1
2 −δ
T <γc ≤2T
(βc − b)
X 1
βc −
2
1
βc < 2 −δ
T <γc ≤2T
.
The second sum on the right is bounded by ǫT log T by Theorem 4.6. Since
any term in the first sum on the right is < −δ, we obtain
X
1
1
c
−δN−
− δ, T ≥
(βc − b) −
− b N c (T ) + O(ǫT log T ).
2
2
T <γ ≤2T
c
In view of Theorem 4.7 and Corollary 4.8 we get
1
ǫ
c
N−
− δ, T ≪ T log T.
2
δ
Putting ǫ = δ 2 we obtain the assertion of the theorem. •
158
Chapter 4
The Selberg class
Thus, subject to the truth of the Lindelöf hypothesis, we get by comparing
Corollary 4.8 and Theorem 4.10, for any positive ǫ,
1
1
c
c
− ǫ, T + N+
+ ǫ, T ≪ ǫN c (T ),
N−
2
2
so the c-values are clustered around the critical line for any c. The distribution
of the c-values close to the real axis is quite regular. It can be shown that
there is always a c-value in a neighbourhood of any trivial zero of L(s) with
sufficiently large negative real part, and with finitely many exceptions there
are no other in the left half-plane. The main ingredients for the proof are
Rouché’s theorem, Lemma 2.19, and Stirling’s formula (1.53). Consequently,
with regard to (4.1), the number of these c-values having real part in [−R, 0]
is asymptotically 21 dL R. On the other side, by (4.8) the behaviour nearby
the positive real axis is very regular. Note that all results from above hold
as well with respect to c-values from the lower half-plane.
Now let NLc (σ, T ) count the number of c-values ρc = βc + iγc of L(s) satisfying σ ≤ βc ≤ 1, |γc | ≤ T . Using Corollary 4.8 with 2−n T for n ∈ N instead
of T and adding up, we get, for fixed σ ≤ 0,
∞
X
NLc (σ, T ) = 2
N c (σ, 2−n T )
n=1
=
X
∞
T
T
dL
1
2
T log + log(λQ )
π
e
π
2n
n=1
∞
dL X log 4 − n log 2
+ O(log T ).
+ T
π n=1
2n
The appearing infinite series are equal to 1 and 0, respectively. Hence, this
summation removes the factor 4 in the logarithmic term, and we have proved
˜ For any σ ≤ 0 and any complex c 6= 1,
Theorem 4.11. Let L ∈ S.
dL
T
T
NLc (σ, T ) =
T log + log(λQ2 ) + O(log T ).
π
e
π
The case c = σ = 0 (the nontrivial zeros of L(s)) is a precise Riemann-von
Mangoldt formula (1.12). Similar results were obtained by Perelli [166] and
Lekkerkerker [129] for other classes of Dirichlet series.
In the exceptional case c = 1 one has to consider the function
qs
(L(s) − 1),
ℓ(s) =
a(q)
where q is the smallest integer greater than one such that a(q) 6= 0. Then,
by a similar reasoning as in the proof of Theorem 4.11, one gets analogous
results. For the special case of the zeta-function this is carried out in Steuding
[193, 194] where Levinson’s method is applied to Epstein zeta-functions.
Section 4.3
The Riemann–von Mangoldt formula
159
4.3.4. Some related results of Selberg. We conclude with some results from Selberg [180]. Under assumption of the truth of the Riemann
hypothesis he obtained for c 6= 1 the asymptotic formula
√
X nL p
1
=
log log T
βc −
3 T
2
4π 2
βc > 1
2
0<γc <T
|c|
(log log log T )3
T
+O T √
+ log
,
4π
1 − |c|2
log log T
where nL is the quantity appearing in Selberg’s Conjecture A. Furthermore,
for
√
dL ν
log log T
1
and
ξ := √
σ(T ) := − ν
2
log T
2 πnL
with positive ν, he proved
X
(βc − σ(T ))
βc >σ(T )
0<γc <T
Z ∞
p
exp(−πξ 2 )
2
+ξ−ξ
exp(−πx ) dx T log log T
2π
ξ
Z ∞
T
2
+ log |c|
exp(−πx ) dx − log |1 − c|
2π
ξ
(log log log T )3
+O T √
.
log log T
1
=
2
r
nL
π
From these results Selberg deduced that about half of the c-values lie to
the
√ left of the critical line, statistically well distributed at distances of order
log log T
off σ = 12 , and that
log T
Z ∞
c
c
NL (σ(T ), T ) ∼ NL (T )
exp(−πx2 ) dx.
−ξ
Most of the remaining c-values lie rather close to the critical line at distances
(log log log T )3
√
of order not exceeding log
. This improves some results due to Selberg
T log log T
(unpublished) and Joyner [104] and gives a much more detailed description
of the clustering of the c-values around the critical line.
Exercise 96. * Prove Lemma 4.2. Read Potter [171] and understand the proof
of Theorem 4.4. Deduce Corollary 4.5.
Exercise 97. * Prove similar estimates for the c = 1-values of L(s) ∈ S.
160
Chapter 4
The Selberg class
4.4. Primitivity and Selberg’s conjectures
The Selberg class is multiplicatively closed. Therefore it makes sense to
introduce the notion of primitive elements. A function L ∈ S is called primitive if it cannot be factored as a product of two elements non-trivially, i.e.,
the equation
L = L1 L2
with L1 , L2 ∈ S
implies L = L1 or L = L2 . This definition of primitivity (analogously to the
one in algebra) is very natural and useful for studies of the structure of the
Selberg class.
4.4.1. Factorization into primitive functions. The ring of integers
is a unique factorization domain: any integer has (up to order) a unique
factorization into powers of prime numbers. Something similar can be shown
for the Selberg class. Conrey & Ghosh [38] proved
Theorem 4.12. Every function in the Selberg class has a factorization into
primitive functions.
Proof. Suppose that L is not primitive, then there exist functions L1 and
L2 in S \ {1} such that L = L1 L2 . Taking into account (4.2) we have
NL (T ) = NL1 (T ) + NL2 (T ),
resp. for the according degrees
dL = dL1 + dL2 .
In view of Theorem 4.1 both L1 and L2 have degree at least 1. Thus, each of
dL1 and dL2 is strictly less than dL . A continuation of this process terminates
since the number of factors is ≤ dL , which proves the claim. •
In connection with Theorem 4.1 it follows that any element of the Selberg
class of degree one is primitive; e.g., Riemann’s zeta-function and Dirichlet L-functions attached to primitive characters. A more advanced example
of primitive elements are L-functions associated with newforms due to M.R.
Murty [156] and further examples were given by Molteni & Steuding [148] by
L-functions to modular forms of Hecke groups. On the contrary, Dedekind
zeta-functions to cyclotomic fields 6= Q are not primitive. In the following section we will consider whether factorization into primitive elements is
unique.
Section 4.4
Primitivity and Selberg’s conjectures
161
4.4.2. Selberg’s conjectures. Denote by aL (n) the coefficients of the
Dirichlet series representation of L ∈ S. The central claim concerning primitive functions is part of
Selberg’s conjectures.
A) For all 1 6= L ∈ S there exists a positive integer nL such that
X |aL (p)|2
p≤x
p
= nL log log x + O(1);
B) for any primitive functions L1 and L2 ,
X aL (p)aL (p) log log x + O(1) if L1 = L2 ,
1
2
=
O(1)
otherwise.
p
p≤x
In some sense, primitive functions are expected to form an orthonormal system.
In view of the factorization into primitive functions, Theorem 4.12, it is
easily seen that Conjecture B implies Conjecture A. In some particular cases
it is not too difficult to verify Selberg conjecture A. For instance, ζ(s) satisfies Selberg’s Conjecture A (see Chapter 1) and, obviously, the same holds
for Dirichlet L-functions too. Liu, Wang & Ye [138] proved Selberg’s Conjecture B for automorphic L-functions L(s, π) and L(s, π ′ ), where π and π ′ are
automorphic irreducible cuspidal representations of GLm (Q) and GLm′ (Q), respectively (we shall give a rough definition of these objects in a later section);
their result holds unconditionally for m, m′ ≤ 4 and in other cases under the
assumption of the convergence of
X |aπ (pk )|2
p
pk
(log p)2
for k ≥ 2, where aπ (n) denote the Dirichlet series coefficients of L(s, π). The
latter hypothesis is an immediate consequence of the Ramanujan hypothesis.
We return to the theme of factorization into primitive elements. Conrey
and Ghosh [38] proved
Theorem 4.13. Selberg’s conjecture B implies that every L ∈ S has a unique
factorization into primitive functions.
Proof. Suppose that L has two factorizations into primitive functions:
L=
m
Y
j=1
Lj =
n
Y
k=1
L̃k ,
Chapter 4
162
The Selberg class
and assume that no L̃k is equal to L1 . Then it follows from
m
n
X
X
aLj (p) =
aL̃k (p)
j=1
that
k=1
m X
X
aLj (p)aL1 (p)
j=1 p≤x
p
n X
X
aL̃k (p)aL1 (p)
=
.
p
k=1 p≤x
By Selberg’s conjecture B, the left-hand side tends to infinity for x → ∞,
whereas the right-hand side is bounded, giving the desired contradiction. •
4.4.3. Prime number theorems. The Selberg conjectures refer to the
analytic behaviour at the edge of the critical strip. Conrey & Ghosh [38]
proved the non-vanishing on the line σ = 1 subject to the truth of Selberg’s
Conjecture B:
Theorem 4.14. Let L ∈ S. If Selberg’s Conjecture B is true, then
L(s) 6= 0
for
σ ≥ 1.
It is conjectured that the Selberg class consists only of automorphic Lfunctions, and for those Jacquet & Shalika [103] obtained an unconditional
non-vanishing theorem.
Proof of Theorem 4.14. In view of the Euler product representation in the
half-plane σ ≥ 1 zeros can only occur on the line σ = 1. By Theorem 4.12
it suffices to consider primitive functions L ∈ S. In case of ζ(s) it is known
that there are no zeros on σ = 1 (see Chapter 1.4.1). It is easily seen that if
Selberg’s conjecture B is true and if L ∈ S has a pole at s = 1 of order m,
then the quotient L(s)/ζ(s)m is an entire function. Hence we may assume
that L(s) is entire. Then L(s + iα) is for any real α a primitive element of
S. Selberg’s Conjecture B applied to L(s + iα) and ζ(s) yields
X aL (p)
(4.17)
≪ 1.
p1+iα
p≤x
Now suppose that L(1 + iα) = 0. Then
L(s) ∼ c(s − (1 + iα))k
as s = σ + iα → 1 + iα for some complex c 6= 0 and some positive integer k.
It follows that
(4.18)
as σ → 1+. Since
log L(σ + iα) ∼ k log(σ − 1)
log L(s) =
X aL (p)
p
ps
+ O(1)
Section 4.4
Primitivity and Selberg’s conjectures
163
for σ > 1, we get by partial summation
Z ∞X
X aL (p)
aL (p) dx
=
(σ
−
1)
.
log L(σ + iα) ∼
σ+iα
1+iα xσ
p
p
1
p
p≤x
By (4.17) the right-hand side is bounded as σ → 1+, which contradicts (4.18).
The theorem is proved. •
As we have seen in Chapter 1, the non-vanishing of L-functions on the
edge of the critical strip is closely related to prime number theorems (here
we mean asymptotic formulae for the Dirichlet coefficients). Indeed, if an
element of the Selberg class L(s) has no zeros in the half-plane σ ≥ 1, we
shall expect the asymptotic formula
X
(4.19)
ψL (x) :=
ΛL (n) = kL x + o(x),
n≤x
where kL = 0 if L(s) is regular at s = 1, otherwise kL is the order of the pole
of L(s) at s = 1, and ΛL (n) is the von Mangoldt-function, defined by
∞
X
L′
ΛL (n)
− (s) =
.
s
L
n
n=1
As a matter of fact, we shall even expect that (4.19) is equivalent to the nonvanishing of L(s) on the 1-line. It is not too difficult to verify this statement
(by application of a Tauberian theorem) for polynomial Euler products in
the Selberg class, i.e.,
−1
m YY
αj (p)
(4.20)
L(s) =
1−
,
ps
p j=1
where m is a fixed positive integer and for each prime p and 1 ≤ j ≤ m the
αj (p) are certain complex numbers (it is easily seen that they have absolute
value less than or equal to one subject to the Ramanujan hypothesis). In
view of Theorem 4.14 it follows that
Corollary 4.15. Assume Selberg’s Conjecture B. The prime number theorem
(4.19) holds for elements of the Selberg class of the form (4.20).
However, Conjecture B might be a rather strong condition if we are interested
in a prime number theorem for a single L-function. Recently, Kaczorowski
& Perelli [108] obtained a more satisfying condition. For this aim they introduced a weak form of Selberg’s Conjecture A:
Normality conjecture. For all 1 6= L ∈ S there exists a non-negative
integer kL such that
X |aL (p)|2
= kL log log x + o(log log x).
p
p≤x
Chapter 4
164
The Selberg class
Assuming this hypothesis, they proved the claim of Theorem 4.14, namely
the non-vanishing of any L(s) on the line σ = 1, and that this statement is
equivalent to the asymptotic formula (4.19). It should be noted that their
proof of L(1+iR) 6= 0 for a given L involves the assumption of their normality
conjecture for several elements in S. In fact, their proof relies on a density
theorem for S (generalizing our approach from Chapter 1.11). Let NL (σ, T )
count the number of zeros ρ = β + iγ of L(s) with β > σ and |γ| < T
(counting multiplicities). Then Kaczorowski & Perelli [108] proved that
uniformly for
close to 1.
1
2
NL (σ, T ) ≪ T 4( dL +3)(1−σ)+ǫ
≤ σ ≤ 1. Unfortunately, this estimate is only useful for σ
4.4.4. Pair correlation in the Selberg class. Assuming the truth
of the Riemann hypothesis Montgomery [149] studied the distribution of
consecutive zeros 21 + iγ, 21 + iγ ′ of the Riemann zeta-function. Montgomery’s
famous pair correlation conjecture states that, for fixed α, β satisfying 0 <
α < β,
1
(γ − γ ′ ) log T
′
lim
♯ 0 < γ, γ < T : α ≤
≤β
T →∞ N(T )
2π
2 !
Z β
sin πu
(4.21)
du.
=
1−
πu
α
Montgomery claims that (4.21) would follow from a sufficiently good estimate
for
X
Λ(n)Λ(n + h) − c(h)x
n≤x
in a certain range of h, where c(h) is some quantity depending on h; however,
the Hardy-Littlewood twin prime conjecture [80] is too strong for an input
into this problem. The pair correlation conjecture has many important consequences; e.g., (4.21) implies that almost all zeros of the zeta-function are
simple.
Dyson remarked shortly afterwards that the function on the right of (4.21)
is the pair correlation function of the eigenvalues of large random Hermitian
matrices, or more specifically of the Gaussian Unitary Ensemble. This supports an old idea of Hilbert and Pólya. Their approach towards Riemann’s
hypothesis was to look for a self-adjoint Hermitian operator whose eigenvalues are ρ − 21 where ρ is a nontrivial zero of the zeta-function; then the
property of being self-adjoint would imply that all zeros ρ lie on the critical
line σ = 21 . In the last years big progress in this direction was made. By the
work of Odlyzko [161] it turned out that the pair correlation and the nearest
neighbour spacing for the zeros of ζ(s) were amazingly close to those for the
Section 4.4
Primitivity and Selberg’s conjectures
165
Gaussian Unitary Ensemble. There is even more evidence for the pair correlation conjecture than numerical data. In the meantime many results from
random matrix theory were found which fit perfectly to certain results on the
value-distribution of the Riemann zeta-function (and even other L-functions;
see Conrey’s survey article [37]).
For example, Keating & Snaith showed that certain Random Matrix ensembles have in a sense the same value-distribution as the zeta-function on
the critical line predicted by Selberg’s limit law. More precisely, Keating &
Snaith [114] showed for characteristic polynomials ZN (θ, U) of the Circular
Unitary Ensemble U(N) the limit theorem:
)
(
log ZN (θ; U)
∈R
lim meas U ∈ U(N) : q
N →∞
1
log
N
2
ZZ
1
exp − 12 (x2 + y 2 ) dxdy,
=
2π
R
where R is any rectangle in the complex plane with edges parallel to the realand the imaginary axis. For the zeta-function there is an old result of Selberg
(unpulished) showing the same Gaussian normal distribution:
)
(
log ζ( 12 + it)
1
∈R
lim meas t ∈ [T, 2T ] : q
T →∞ T
1
log
log
T
2
ZZ
1
exp − 12 (x2 + y 2 ) dxdy.
=
2π
R
The first published proof of the latter result is due to Joyner [104].
Further evidence for the pair correlation conjecture was discovered by Rudnick & Sarnak. Normalize the ordered nontrivial zeros ρn = 21 +iγn by setting
γn
γ̃n =
log |γn |,
2π
then it follows from the Riemann-von Mangoldt formula (1.12) that the numbers γ̃ have unit mean spacing. Then the pair correlation conjecture (4.21)
can be rewritten as follows: for any nice function f on (0, ∞)
Z ∞
X
lim
f (γ̃n+1 − γ̃n ) =
f (x)P (x) dx
N →∞
n≤N
0
where P is the distribution of consecutive spacings of the eigenvalues of a large
random Hermitean matrix. Rudnick & Sarnak [177] succeeded in showing
that the m-dimensional analogue of the latter formula, the m-level correlation, holds for a large class of test functions. Finally, note that Katz &
Sarnak [113] proved a function field analogue of Montgomery’s pair correlation conjecture without assuming any unproved hypothesis.
166
Chapter 4
The Selberg class
Recently, Murty & Perelli [158] extended Montgomery’s argument to the
Selberg class. For this purpose they considered two primitive functions L1
and L2 from S. To compare the zeros 21 + iγL1 of L1 against the zeros 12 + iγL2
of L2 define
X
π
T iα dL1 (γL1 −γL2 ) w(γL1 − γL2 ),
F (α; L1, L2 ) =
dL1 T log T −T ≤γ ,γ ≤T
L1
L2
where w is a suitable weight function. The pair correlation conjecture for the
Selberg class takes then the form:
Pair correlation conjecture. Let L1 and L2 be primitive functions in S.
Under the assumption of the Grand Riemann hypothesis, uniformly in α, as
T → ∞,
δL1 ,L2 |α| + dL1 T −2|α| dL1 log T (1 + o(1)) if |α| < 1 ,
F (α; L1, L2 ) ∼
δL1 ,L2 |α|
otherwise.
Here
1 if L1 = L2 ,
0 otherwise,
is the Kronecker-symbol. The general pair correlation conjecture includes
Montgomery’s pair correlation conjecture. It has plenty of important applications as M.R. Murty & Perelli [158] worked out (for instance, the Artin
conjecture follows from the pair correlation conjecture). The pair correlation
conjecture implies that almost all zeros of two primitive functions L1 and L2
are simple and distinct. Moreover, if the pair correlation formula holds for at
least one value of α, then S has unique factorization into primitive functions.
This shows what a powerful tool the pair correlation is. Further, M.R. Murty
& Perelli proved
δL1 ,L2 :=
Theorem 4.16. The Grand Riemann hypothesis and the pair correlation
conjecture imply the Selberg conjectures.
The pair correlation conjecture plays a complementary role to the Riemann
hypothesis: vertical vs. horizontal distribution of the nontrivial zeros of ζ(s).
Both together seem to be the key to several unsolved problems in number
theory!
Exercise 98. i) Prove that Selberg’s Conjecture B implies Conjecture A
ii) Show that Selberg’s Conjecture B holds for pairs of Dirichlet L-functions.
Exercise 99. If the Selberg conjecture B is true, L ∈ S is primitive if and only if
nL = 1, where the quantity nL from Selberg’s Conjecture A.
Section 4.5
Hecke L-functions
167
Exercise 100. Assuming Selberg’s conjecture B, prove that if L ∈ S has a pole at
s = 1 of order m, then the quotient L(s)/ζ(s)m is an entire function.
Exercise 101. Show that a polynomial Euler product of the form (4.20) satisfies the Ramanujan hypothesis and, conversely, that the Ramanujan hypothesis for
(4.20) implies that |αj (p)| ≤ 1.
Exercise 102. * Prove Corollary 4.15: Assuming Selberg’s Conjecture B, given
a polynomial Euler product (4.20) in the Selberg class, prove the prime number
theorem (4.19).
Hint: apply the Tauberian theorem of Wiener-Ikehara 1.14.
Exercise 103. * Suppose that L1 , L2 ∈ S and the Dirichlet coefficients of both
Dirichlet series are equal for all but finitely many prime numbers: aL1 (p) = aL2 (p).
Assuming Selberg’s Conjecture B, show that L1 = L2 .
4.5. Hecke L-functions
In 1920, Hecke [87] introduced a new class of L-functions which generalize
the concepts of Dedekind zeta-functions and Dirichlet L-functions.
Let K be a number field, f be an ideal of K, and χ modulo f be a grössencharacter (the definition will be given in the following subsection). Then the
associated Hecke L-function is given by
−1
X χ(a)
Y
χ(p)
L(s, χ) =
=
1−
(4.22)
,
s
s
N(a)
N(p)
a
p
where the sum is taken over all non-zero integral ideals a of K, the product
is taken over all prime ideals p, and N(a) denotes the norm of the ideal a;
the identity between the Dirichlet series and the Euler product follows from
the unique prime ideal factorization.
4.5.1. Grössencharacters. Hecke grössencharacters represent the most
general extension of Dirichlet characters to number fields.
Given a number field K of degree n over Q, there are exactly n automorphisms K(j) of K into C, for 1 ≤ j ≤ n, given by
K ∋ α 7→ α(j) ∈ K(j) ,
where the α(j) denote the conjugates of α; we assume that among these there
are r1 real and 2r2 complex embeddings (that makes n = r1 +2r2 ). We denote
the real embeddings by
K(1) , . . . , K(r1 )
and the complex embeddings which are pairwise complex conjugate by
K(r1 +1) , . . . , K(r1 +r2 +1) = K(r1 +1) , . . . , K(n) = K(r1 +r2 ) .
Chapter 4
168
The Selberg class
Let f be a non-zero integral ideal of K. The unit group modulo f is defined
to be the set of all units ǫ ≡ 1 mod f which are totally positive and we denote
it by U(f). It is easily seen that U(f) is a group. By Dirichlet’s unit theorem
there exist r = r1 + r2 − 1 units η1 , . . . , ηr and a root of unity ζ in K such
that any ǫ ∈ U(f) has a unique representation
ǫ = ζ m η1n1 · . . . · ηrnr
with integers m, nk . The units η1 , . . . , ηr are said to be fundamental units of
U(f) although they are not uniquely determined. Define the matrix
1
if 1 ≤ j ≤ r1 ,
(j)
(ej log |ηk |)1≤j,k≤r ,
where ej :=
2 if r1 < j ≤ r1 + r2 .
Then the regulator R(f) is defined to be the absolute value of the determinant
of this matrix:
(j)
R(f) = | det(ej log |ηk |)|;
it should be noticed that the regulator does not depend on the choice of the
fundamental units ηk .
We further denote by I(f) the multiplicative group generated by all ideals
coprime with f. The principal ray class P(f) is the subgroup of I(f) consisting
of all principal ideals of the form (α/β) satisfying
• 0 6= α, β ∈ OK (the ring of integers);
• α ≡ β mod f;
• α/β is totally positive, i.e., all its real conjugates are positive.
The factor group
G(f) := I(f)/P(f)
is called the ray class group mod f, and its elements are called ray classes.
The ray classes are the analogues of the residue classes in the rational number
field case. One can show that G(f) is a finite abelian group and we denote its
order by h(f).
We shall give a brief
√ example. For the sake of simplicity we shall consider
the number field Q( −5) and choose f = (1) in which case G((1)) is the class
group. We have already seen in Chapter 1.1.5.3 that there is no unique prime
factorization, and so the class number h = h((1)) is greater than one. One
can deduce from Minkowski’s theorem on linear forms that every class of G
contains an integral ideal a with norm
p
N(a) ≤ | dK |,
where dK is the discriminant of the number field, that is in our case dK =
−20. Obviously, this observation proves the finiteness of the class number.
However, we can also use it to get an overview of the structure of the class
group. For this purpose we observe that the only prime ideals p with N(p) ≤ 4
Section 4.5
Hecke L-functions
169
can be among the prime ideal divisors of (2) and (3). By the splitting of
primes in quadratic number fields (see again Chapter 1.1.5.3), we find
√
√
(2) = p21
with p1 = (2, 1 + −5) = (2, 1 − −5),
√
√
(3) = p2 p′2
with p2 = (3, 1 + −5) 6= p′2 = (3, 1 − −5).
Hence, the ideals with norms less than or equal to 4 are p1 , p2 , p′2 , and (2) =
p21 . It is easy to see that p1 is not principal and represents a class of order
two. Furthermore, it is easy to see that all other ideals lie in this class or
√
are principal. Hence the class number of Q( −5) is two and we have a
description of the associated class group.
Now we are in the position to define Hecke characters. Suppose we are
given numbers aj and νk satisfying
• aj ∈ {0, 1} for 1 ≤ j ≤ r1 and aj ∈ Z for r1 < j ≤ r1 + r2 ;
• νk ∈ R for 1 ≤ k ≤ r1 + r2 such that ν1 + . . . + νr1 +r2 = 0.
Then we define a function χ∞ : K∗ → C∗ by
χ∞ (α) =
rY
1 +r2
k=1
(k) iνk
|α |
rY
1 +r2
j=1
α(j)
|α(j)|
aj
.
Obviously, χ∞ is unimodular. Since the sum of the νk vanishes, it follows
that χ∞ is trivial on Q∗ . We suppose that the kernel of χ∞ contains the
unit group modulo f, i.e., χ∞ (ǫ) = 1 for any ǫ ∈ U(f). Then χ∞ induces a
character on P(f).
If a non-trivial homomorphism χ : I(f) → C∗ is identified with χ∞ on P(f),
that is
χ(a) = χ∞ (α)
for a = (α) ∈ P(f),
then χ is said to be a grössencharacter modulo f (resp. Hecke character in
some literature). If all numbers aj , νk are equal to zero, then χ is said to
be a ray class character, and if additionally f = (1), then χ is an ideal class
group character (that is one of the finitely many characters of the class group
of K). If there exists an ideal f∗ ⊂ f and a grössencharacter χ∗ mod f∗ such
that χ = χ∗ on I(f), then χ is said to be induced by χ∗ ; otherwise χ is called
primitive and f is said to be the conductor of χ. (For details from algebraic
number theory we refer to Narkiewicz [159].)
4.5.2. Analytic properties and arithmetic consequences. Now we
return to the associated Hecke L-functions to number fields K. Given a
grössencharacter χ modulo f, we extend χ to the group I of all fractional
ideals of K by setting χ(a) = 0 if a is not coprime with f. Then we may
define (formally) the Dirichlet series and the Euler product appearing in
(4.22). Hecke L-functions to grössencharacters are the analogues of Dirichlet
Chapter 4
170
The Selberg class
L-functions: if K = Q, f = (q) with q ∈ Z, and χ∞ ≡ 1, then the construction
above leads without the totally positive condition to
G(f) = (Z/qZ)∗ /{±1}.
If χ is the trivial (principal) character, then the Hecke L-function for a number field K is nothing but the Dedekind zeta-function. Notice that what we
call Hecke L-functions are in some literature called (generalized) Dirichlet
L-functions.
Both the series and the product (4.22) defining L(s, χ) are absolutely convergent for σ > 1 and uniformly in any compact subset. To see this we
recall from class field theory that in a number field K of degree n over Q any
rational prime number p has a unique factorization into a product of prime
ideals
r
r
Y
X
ej
fj
pj
with N(pj ) = p
and
ej fj = n;
(4.23)
(p) =
j=1
j=1
of course, the integers ej , fj , and r depend on p (which is not indicated here
for simplicity). Hence we can rewrite (4.22) as an ordinary Euler product
−1
−1 Y Y
r Y
χ(pj )
χ(p)
.
=
1 − sfj
L(s, χ) =
1−
N(p)s
p
j=1
p
p
pj |(p)
Thus, L(s, χ) has a representation as a polynomial Euler product. Hence we
may also rewrite this as an ordinary Dirichlet series:
∞
X
a(n)
,
L(s, χ) =
ns
n=1
where
a(n) =
Y
p|n
X
k1 ,...,kr ≥0
k1 f1 +...+kr fr =ν(n;p)
r
Y
χ(pj )kj .
j=1
Since the degree of the local Euler factors is bounded by the degree of the
field extension K/Q, it immediately follows that the Ramanujan hypothesis
holds.
In 1920, Hecke [87] proved that L(s, χ) extends to an entire function and
satisfies a functional equation of Riemann-type provided χ is primitive. Let
dK denote the discriminant of K. We define
1
rY
1 +r2
iνk
| dK |N(f) 2 −r2
2 ,
2 2 , A(f) =
γ(χ) =
πn
k=r +1
1
and
rY
r1
1 +r2
Y
s + aj − iνj
|aj | − iνj
Γ(s, χ) =
Γ
.
Γ s+
2
2
j=1
j=r +1
1
Section 4.5
Hecke L-functions
171
Then
Λ(1 − s, χ) = ω(χ)Λ(s, χ),
where ω(χ) is a complex number with |ω(χ)| = 1, depending only on χ, and
Λ(s, χ) := γ(χ)A(f)s Γ(s, χ)L(s, χ).
Hecke’s proof of the functional equation is rather complicated; modern proofs
use Tate’s approach via harmonic analysis (see Chapter 2.1.6). In view of all
the mentioned properties it follows that Hecke L-functions L(s, χ) to primitive grössencharacters are elements of the Selberg class of degree n = [K : Q].
The Hecke L-function to the trivial character is, as already mentioned, equal
the Dedekind zeta-function and so it is an element of the Selberg class too.
We sketch some arithmetic consequences of the analytic properties of Hecke
L-functions; most of the details can be proved in a similar way as in Chapter 1
(for the distribution of primes in arithmetic progressions or in our applications
of Tauberian theorems). First of all, we notice that L(s, χ) does not vanish
on the edge of the critical strip:
(4.24)
L(1 + it, χ) 6= 0
for t ∈ R.
We have already mentioned that L(s, χ) is entire and so it is regular at s = 1
unless χ is trivial. If χ is not trivial, then
X
x
χ(p) = o
log x
N(p)≤x
as x → ∞; for trivial χ we have the prime ideal theorem 1.16. This information might be used to verify Selberg’s Conjecture A for Hecke L-functions.
First of all we note that
r
X
a(p) =
χ(pj ).
j=1
fj =1
Here the summation is taken over the prime ideals of degree one lying above
p; however, this condition is negligible if we deal with
r
X |a(p)|2 X X
|χ(pj )|2
=
+ O(1),
p
p
p≤x
p≤x j=1
where we used the orthogonality relations for characters in the last step. Now
the asymptotics of Conjecture A follow by partial summation from (4.25).
In analogy to our studies on the distribution of the prime numbers in prime
residue classes we shall now decompose the ray class group G(f) into its ray
classes C. Here an application of the Tauberian theorem 1.14 leads to
1
π(x).
(4.25)
♯{p ∈ C : N(p) ≤ x} ∼
♯G(f)
Chapter 4
172
The Selberg class
√
Exercise 104. Describe
√ the class group of Q( 10) and the ray class groups mod
√
(2) of Q( −5) and Q( 10).
Exercise 105. Let K be a quadratic number field and σ the nontrivial element of
the Galois group Gal(K/Q). Then K∗ /Q∗ ≈ {α ∈ K∗ : N(α) = 1} by the map
α 7→ α/ασ . Show that if K is imaginary, then
ν
α
χ∞ (α) =
|α|
for some integer ν, and if K is real, then
a1 σ a2
α
α
χ∞ (α) =
|α|
|ασ |
for some a1 , a2 ∈ {0, 1}.
Exercise 106. Give a detailed proof for Selberg’s Conjectures A and B for Hecke
L-functions.
Exercise 107. * Show (4.24) and prove the asymptotic formula (4.25).
Hint: apply the Tauberian theorem of Wiener-Ikehara 1.14.
Exercise 108. * Prove a non-trivial zero-free region for Hecke L-functions and
improve the statement of the previous exercise by giving an explicit error estimate
in (4.25).
The nest exercise deals with the Gaussian field Q(i). Recall from Chapter 1.5.3
that every ideal of the Gaussian ring of integers Z[i] is principle and that the
Gaussian primes π are given by π = a + bi with N(π) = a‘2 + b2 = p for some
prime number p ≡ 1 mod 4.
Exercise 109. ** i) Show that the function
4im
α
= exp(4im arg(α))
a = (α) 7→ χm (a) =
|α|
for a 6= 0 and any integer m is a primitive grössencharacter.
ii) Prove that the associated Hecke L-function L(s, χm ) satisfies the functional
equation
π −s Γ(s + 2|m|)L(s, χm ) = π s−1 Γ(1 − s + 2|m|)L(1 − s, χm ).
iii) Deduce from the prime number theorem that
X π 4im
= δm (1 + o(1))Li (x),
|π|
|π|≤x
where δm is equal to 1 if m = 0 and equal to 0 otherwise. Furthermore, show that
the Gaussian primes are equidistributed in sectors:
β−α
♯{π ∈ Z[i] : |π| ≤ x, α < arg π < β} ∼
Li (x).
π
Exercise 110. Show that the L-function LE1 (s) attached to the elliptic curve
E1 from Chapter 3.6.3 is indeed a Hecke L-function to a grössencharacter χ of
√
Q( −1).
Section 4.6
Artin L-functions
173
4.6. Artin L-functions and Artin’s conjecture
Now we want to study a further class of L-functions which play a central
role in algebraic number theory ever since Artin introduced them in order to
find higher reciprocity laws. However, first of all we shall briefly motivate
their definition.
4.6.1. A fundamental problem in number theory. In algebraic
number theory, a fundamental problem is to describe how a rational prime
factors into primes in the ring of integers OK of an arbitrary number field
K. Now assume that K is a Galois extension over Q with Galois group
G := Gal(K/Q) (i.e., Q is fixed with respect to automorphisms from G).
Then K is the splitting field of some monic polynomial with rational coefficients, and G is the group of field automorphisms of K fixing Q pointwise.
The splitting type of p in OK is completely determined by the size of the
subgroup of G which fixes any pj . For simplicity, assume that the rational
prime p is unramified in K, i.e., the primes pj in (4.23) are all distinct, then
these subgroups are all cyclic. Information about the factorization of such p
is encoded in the so-called Frobenius automorphism σpj of G, the canonical
generator of the subgroup of G which maps any pj into itself. The Frobenius
is determined only up to conjugacy in G; nevertheless, the resulting conjugacy class, which we denote by σp , completely determines the splitting type
of (4.23).
If, for example,
K = Q(i) = {a + bi : a, b ∈ Q},
then
OK = Z[i] = {m + ni : m, n ∈ Z}.
In this case, σp is the identity if −1 is a quadratic residue mod p, and the
complex conjugation otherwise. Hence, we may identify G with the subgroup
{±1} of C∗ := C \ {0} via the homomorphism ρ : G → {±1}:
−1
.
ρ(σp ) =
p
By a part of the quadratic reciprocity law, the Legendre symbol can be
expressed in terms of a congruence condition on p which states for unramified
(odd) primes p
p−1
−1
+1 if p ≡ 1 mod 4,
= (−1) 4 =
−1 otherwise.
p
Thus, the factorization of p in Z[i] depends only on its residue mod 4 (see
also Chapter 1.5.3).
174
Chapter 4
The Selberg class
One goal of class field theory is to find a similar description of σp for
arbitrary Galois extensions K. In general, one cannot expect that there
exists a modulus q such that σp is the identity if and only if p lies in some
arithmetic progression mod q. However, if K is abelian, i.e., G = Gal(K/Q)
is abelian, and ρ : G → C∗ is a homomorphism, then it is known that there
exists a Dirichlet character ψ mod q such that
(4.26)
ψ : (Z/qZ)∗ → C∗
with ρ(σp ) = ψ(p)
for all primes p, unramified in K. This is a reformulation of the famous
Kronecker-Weber theorem (stating that any finite abelian extension of Q is
)). It follows that the splitting
contained in some cyclotomic field Q(exp( 2πi
n
properties of p in K depend only on its residue modulo some fixed number q
depending on K. In particular, this implies the general quadratic reciprocity
law of Gauss. As a matter of fact, the factorization of Dedekind zeta-functions
p
ζK (s) = ζ(s)L(s, χ)
with K = Q( χ(−1)q)
for all quadratic fields K is equivalent with quadratic reciprocity. Artin’s
reciprocity law of abelian class field theory gives an extension of (4.26) for
abelian fields.
What can be said for nonabelian Galois extensions? Recognizing the utility
of studying groups in terms of their matrix representations, Artin focused
attention on homomorphisms
ρ : G = Gal(K/Q) → GLm (C),
i.e., m-dimensional representations of the Galois group G; note that onedimensional representations are simply characters. Artin transferred the
problem of analyzing conjugacy classes in G to the analogous problem in
GLm (C), where the corresponding classes are completely determined by their
characteristic polynomials
ρ(σp )
,
det 1 − s
p
where 1 denotes here the unitary matrix. Introducing the so-called Artin
L-function
−1
Y
ρ(σp )
L(s, ρ) =
det 1 −
ps
p
(we give a precise definition in the following section), Artin was able to reduce
the problem to one involving these analytic objects: is it possible to define
L(s, ρ) in terms of the arithmetic of Q alone? It was in this context that
Artin proved his reciprocity law. Indeed, for abelian K and one-dimensional
ρ, Artin showed that L(s, ρ) is identical to a Dirichlet L-function L(s, ψ) with
Section 4.6
Artin L-functions
175
an appropriate character ψ mod q. Since an identity between two Euler products implies an identity between the local Euler factors (by the uniqueness
of the Dirichlet series expansion), this yields Artin’s reciprocity law.
4.6.2. Artin L-functions. Let L/K be a Galois extension of number
fields with Galois group G. Further, let ρ : G → GLm (V ) be a representation
(group homomorphism) of G on a finite dimensional complex vector space V .
In order to give the definition of the Artin L-function attached to these data,
we recall some facts on prime ideals in number fields and their ramification
in Galois extensions. (For the details from algebraic number theory we refer
once more to Narkiewicz [159].)
For each prime p of K, and a prime P of L with P|p, we define the decomposition group by
DP = {ρ ∈ G : Pρ = P} = Gal(LP /Kp ),
where LP and Kp are the completions of L at P and K at p, respectively.
Denote by kP /kp the residue field extension. By Hensel’s lemma, we have a
surjective map from DP to Gal(kP /kp ); its kernel IP is the inertia group at
P, defined by
IP = {ρ ∈ G : ρ(α) ≡ α mod P for all α ∈ OL }.
We thus have an exact sequence
1 → IP → DP → Gal(kP /kp ) → 1.
Hence, there is an isomorphism
DP /IP ≃ Gal(kP /kp ).
Now kP /kp is a Galois extension of finite fields, and hence the group
Gal(kP /kp ) is cyclic, generated by the map α 7→ αN(p) , where N(p), the absolute norm of p, is the cardinality of kp . We can choose an element σP ∈ DP
whose image in Gal(kP /kp ) is this generator; this σP is called Frobenius element at P, i.e.,
σP (α) ≡ αN(p) mod P
for all α ∈ OL . Note that the Frobenius element is only defined mod IP .
For unramified p (and in particular, these are all but finitely many p), the
Frobenius is well-defined since IP = {1}. The action of the Galois group
on the set of primes in L above p is transitive, and thus for any pair of
primes P1 and P2 lying above p, there exists an automorphism in G which
simultaneously conjugates DP1 into DP2 , IP1 into IP2 , and σP1 into σP2 . This
implies an identity for the characteristic polynomials of σPj on the subspace
VPj of V on which IPj acts trivially:
ρ(σP2 ) ρ(σP1 ) det 1 −
VP1 = det 1 −
VP2 .
N(p)s
N(p)s
Chapter 4
176
The Selberg class
Thus, these characteristic polynomials are independent of the choice of σP .
Denote by σp the conjugacy class of Frobenius elements at primes P above
p; in case of unramified p the inertia group is trivial, and σp is called Artin
symbol.
Following Artin [3], we define the Artin L-function attached to ρ by
−1
Y
ρ(σp ) (4.27)
,
L(s, ρ, L/K) =
det 1 −
VP
N(p)s
p
where p runs through the prime ideals of the ring of integers in K; this Euler
product converges for σ > 1.
“The zeta-function of a field is like the atom of physics. (. . .)
we will show how to split it via group theory.”
This is a quotation of H.M. Stark [188] and in the following section we
illustrate the just given construction by one of his explicit examples.
1
4.6.3. An example. We consider the field K = Q(2 3 ). Notice that K
is not normal over Q (since the polynomial X 3 − 2 has only one of its roots
in K). We write
1
α = 23 ,
β=e
2πi
3
1
23 ,
γ=e
4πi
3
1
23 .
2πi
The field L = Q(α, e 3 ) = Q(α, β, γ) is normal over Q of degree 6. Since
automorphisms of L are determined by their action on α, β and γ, we find
that the Galois group of L is given by
G = Gal(L/Q) = {1, (αβγ), (αγβ), (αβ), (αγ), (βγ)},
which is the symmetric group on three letters. The splitting of primes from
Q to K, and likewise from K to L, is ruled by the Frobenius automorphisms.
Suppose that P is an unramified prime of L which lies above p of K which in
turn lies above the rational prime p of Q. Then the Frobenius automorphism
of P relative to Q is given by one of the following conjugacy classes:
• σP = 1. Since the Frobenius has order one, by (4.23), there are 6
primes in L above p. Obviously, σP ∈ Gal(L/K) = {1, (βγ)}. In this
case p splits in K into three different primes pj (1 ≤ j ≤ 3) each of
which splits into two prime ideals Pk (1 ≤ k ≤ 6) of L.
• σP is in the conjugacy class {(αβ), (αγ), (βγ)} of elements of order
two. We may choose P such that σP = (βγ) ∈ Gal(L/K). The
f = 2 in (4.23) and so there are three second degree primes Pk
(1 ≤ k ≤ 3) above p; we may assume that P = P1 . We observe
that the Frobenius automorphism of P relative to K is equal to
σP . Hence, we find N(P) = N(p)2 and N(p) = p for some prime
p = p1 of K. For the other two primes P2 and P3 the Frobenius
Section 4.6
Artin L-functions
177
σP is equal to (αβ) and (αγ), respectively. In these cases we find
2
σP
= 1 ∈ Gal(L/K) and N(P) = N(p) and N(p) = p2 for some prime
p = p2 of K. Thus, the primes P2 and P3 have relative degree one
over a single prime p2 of K (which is of degree two).
• σP is in the conjugacy class {(αβγ), (αγβ)} of elements of order
three. In this case we have f = 3 in (4.23) and there two third
degree primes P1 and P2 of L above p, for one of them σP = (αβγ)
2
and for the other σP = (αγβ). In both cases neither σP nor σP
lie
in Gal(L/K) = {1, (βγ)} and so both P1 and P2 lie above a single
prime p of K (which must be of degree 3).
Now we want to compute the associated Artin L-functions. First of all we
have a look on every individual Euler factor. Since the field extension K/Q
has degree 3, there are the following possibilities to consider.
• The prime p splits completely into three different prime divisors; e.g.,
(31) = p1 p2 p3 with
p1 = (31, α − 4),
(4.28)
p2 = (31, α − 7),
p3 = (31, α − 20).
In this case the local Euler factor at p is of the form


 −1
−3
1 0 0
1
1
= det 1 −  0 1 0  s  .
1− s
p
p
0 0 1
Obviously, the appearing matrix has the eigenvalue +1 with multiplicity 3.
• The prime p can be factored into a product of two factors, one of
degree one and one of degree two; for example, (5) = p1 p2 with
p1 = (5, α − 3),
p2 = (5, α2 + 3α + 9).
Here we have


−1 −1
0 1
1
1


1− s
1 − 2s
= det 1 −
1 0
p
p
0 0


0 0
(4.29)
= det 1 −  0 1
1 0


1 0


= det 1 −
0 0
0 1
 −1
0
1
0  s
p
1
 −1
1
1
0  s
p
0
 −1
0
1
1  s .
p
0
The eigenvalues of the (similar) matrices are −1 and +1 with multiplicities one and two, respectively.
178
Chapter 4
The Selberg class
• The prime p is a prime ideal of third
case we have


−1
0
1


= det 1 −
0
1 − 3s
p
1


0
(4.30)
= det 1 −  1
0
degree; e.g., (7) = p. In this
 −1
1 0
1
0 1  s
p
0 0
 −1
0 1
1
0 0  s .
p
1 0
Here the eigenvalues of the (similar) matrices are the third roots of
unity.
Before we continue we remark that the splitting of primes can be computed
by the following statement: suppose that g(X) is the minimal polynomial of
α ∈ K over Q and that it splits factors mod p into irreducible pieces as
g(X) ≡ g1 (X)e1 · . . . · gr (X)er mod p.
If the power of p in the polynomial discriminant of g(X) is the same as the
power of p in the relative discriminant DL/K of L/K, then p splits in L as
p = Pe11 · . . . · Perr ,
where Pj = (p, gj (α)) is of relative degree deg gj . This togehter with Eisenstein’s irreducibility criterion gives the basic tools to do arithmetic computations in number fields.
We may represent the Galois group G by matrices as follows. For g ∈ G we
write
 
 
α
α
 β  g = M(g)  β  ,
γ
γ
where M(g) is the permutation matrix corresponding to g. Thus we can
represent the six elements of G by






1 0 0
0 1 0
0 0 1
1 7→  0 1 0  , (αβγ) 7→  0 0 1  , (αγβ) 7→  1 0 0  ,
0 0 1
1 0 0
0 1 0






0 1 0
0 0 1
1 0 0
(αβ) 7→  1 0 0  , (αγ) 7→  0 1 0  , (βγ) 7→  0 0 1  .
0 0 1
1 0 0
0 1 0
The map ρ : g 7→ M(g) defines a homomorphism: M(gh) = M(g)M(h); it is
an example of a three dimensional permutation representation of the group
G. The conjugacy classes of the symmetric group on α, β, γ are precisely the
conjugacy classes of Frobenius automorphisms arising from prime numbers
Section 4.6
Artin L-functions
179
which split in the indicated form and for each of them we observe via (4.28)(4.30) that the associated Euler factors are of the form as predicted by (4.27).
Now we want to introduce a more convenient notation of Artin L-functions.
To any representation ρ of G, we can attach a character χ of G by setting
χ(g) = trace(ρ(g))
for g ∈ G. The degree of a character is defined by deg χ = χ(1). If h is
another element of G, then
ρ(h−1 gh) = ρ(h)−1 ρ(g)ρ(h),
so that ρ(h−1 gh) and ρ(g) are similar matrices and thus have the same trace.
This shows that characters χ of G are constant on the conjugacy classes. Two
representations are said to be equivalent if they have the same character. If
ρ1 and ρ2 are representations of G with characters χ1 and χ2 , then
ρ1 (g)
0
ρ(g) =
0
ρ2 (g)
also defines a representation of G with character χ1 + χ2 , and in this case
ρ is said to be reducible; any representation which is not reducible is called
irreducible. We shall use the same attributes for the associated character.
It turns out that any conjugacy class of G corresponds to an irreducible
representation and one can show that there are not more; of course, distinct
irreducible representations are non-equivalent (these observations are analogous to the case of Dirichlet characters and the group of residue classes of
Z).
In our example we find for the the three conjugacy classes of G:
1
(αβγ), (αγβ) (αβ), (αγ), (βγ)
+1
+1
+1
+1
+1
−1
+2
−1
0
χ0
χ1
χ2
Hence there are three irreducible characters (in some literature simple characters): we are dealing with the trivial character χ0 , another character χ1 of
degree one, and a character χ2 of degree two.
It is easily seen that our characters satisfy the orthogonality relations, that
are
1 X
(♯C)−1
if C = D,
χ(C)χ(D) =
0
otherwise,
♯G
χ∈Ĝ
where C and D are two conjugacy classes, and
1 X
1
if χ = ψ,
ψ(g)χ(g) =
0
otherwise.
♯G
g∈G
Chapter 4
180
The Selberg class
Since the Euler factors in (4.27) depend only on the conjugacy class σp , in
the sequel we will talk sometimes in terms of characters and denote the Artin
L-function (4.27) by L(s, χ, L/K) (and sometimes we shall even write L(s, χ)
for short). To illustrate this we continue with our example. We can construct
more characters from the irreducible characters listed above, for example,
a third degree character χ related to the permutation representation (αβ).
Taking the character relations into account we find χ = χ0 + χ2 . For the
related Artin L-functions we note that
L(s, χ, L/K) = L(s, χ0 + χ2 , L/K) = L(s, χ0 , L/K)L(s, χ2 , L/K).
For the field L = Q(α, β, γ) there are four subfields up
√ to conjugacy. First
of all the field Q itself, fixed by all of G, second Q( −3) fixed by G1 :=
1
{1, (αβγ), (αγβ)}, third K = Q(2 3 ) fixed by G2 := {1, (βγ)}, and finally L
fixed just by {1}.
✉
✉✉
✉✉
✉
✉✉
✉✉
L✹
✹✹
✹✹
✹✹
✹✹
✹✹
✹✹
✹✹
✹✹
✹
1
K = Q(2 3 )
✼✼
✼✼
✼✼
✼✼
✼✼
χ2 ✼✼
✼✼
✼✼
✼✼
√
Q( −3)
Q
✇
✇✇
✇✇
✇
✇✇ χ1
✇✇
{1}
r
rrr
r
r
r
rrr
rrr
G2 ✿
✿✿
✿✿
✿✿
✿✿
✿✿
✿✿
✿✿
✿✿
✿✿
(βγ)
✿✿
✿✿
✿✿
✿✿
✿✿ (αβγ),(αγβ)
✿✿
✿✿
✿✿
✿✿
✿
r
rrr
r
r
rrr
rrr
Gal(L/Q) = S3
We obtain the following factorizations of the associated Dedekind zetafunctions into products of Artin L-functions to L/Q:
ζ(s) = ζQ(s) = L(s, χ0 ),
ζQ(√−3) (s) = L(s, χ0 ) L(s, χ1 ),
ζQ(2 31 ) (s) = L(s, χ0 ) L(s, χ2 ),
ζL (s) = L(s, χ0 ) L(s, χ1 ) L(s, χ2 ).
We observe that any of the Dedekind zeta-functions on the left-hand side is divisible by the Riemann zeta-function (in the sense that their quotient is an entire function). It follows from these factorizations and the analytic behaviour
of Dedekind zeta-functions that each of the involved Artin L-functions with
χ 6= χ0 possesses a meromorphic continuation to the whole complex plane;
the only possible poles can occur at zeros of other Artin L-functions. Furthermore we can deduce functional equations of the Riemann-type. This is a
rather remarkable new way to deduce analytic properties for L-functions!
G1
Section 4.6
Artin L-functions
181
Furthermore, we see that the Dedekind zeta-functions are algebraically
dependent:
ζQ(√−3) (s)ζQ(2 13 ) (s) = ζQ (s)ζK (s).
It is an interesting question to which extent the Dedekind zeta-function
determines the field. One can show that the Dedekind zeta-function ζK (s) determines the minimal normal extension L of Q containing K and thus we have
to ask whether there exist non-conjugate subgroups of Gal(L/Q) giving the
same induced trivial character. This is indeed possible! Two number fields
K1 and K2 are said to be arithmetically equivalent if their Dedekind zetafunctions are the same. The first example of arithmetical equivalent fields
was given by Gassmann [57]. Perlis [167] proved that arithmetically equivalent non-isomorphic fields have at least degree 7 and that this bound cannot
be improved. An explicit example of degree 8 is for instance Q((−3)1/8 ) and
Q((−48)1/8 ) which is due Perlis & Schinzel [168].
4.6.4. The Artin conjecture. One of the most fundamental conjectures in algebraic number theory is
Artin’s Conjecture. Let L/K be a finite Galois extension with Galois group
G. For any irreducible character χ 6= 1 of G the Artin L-function L(s, χ, L/K)
extends to an entire function.
We discuss briefly one of its important consequences. Dedekind’s conjecture
claims that the quotient ζL (s)/ζK(s) is entire provided L/K is an extension
of number fields, not necessarily Galois. If L/K is a Galois extension, then
the so-called Artin-Takagi factorization gives a factorization of the Dedekind
zeta-function of a number field relative to a subfield (see Heilbronn’s survey
[91]); more precisely,
L(s, 1, L/K) = ζK (s),
and L(s, RG , L/K) = ζL (s),
where RG is the regular character of G (the character defined by
and
Y
ζL (s) =
L(s, χ, L/K)χ(1) ,
P
χ
χ(1)χ),
χ∈G̃
where G̃ denotes the set of irreducible characters of G. In case of Galois
extensions L/K, the Aramata-Brauer theorem (see Heilbronn [91] or Murty
& Murty [157], §2.3) yields the truth of Dedekind’s conjecture; its proof
relies mainly on the Artin-Takagi factorization. In the general case, if L/K is
a finite (not necessarily Galois) extension, then Dedekind’s conjecture follows
from Artin’s conjecture by studying the normal closure of L/K.
182
Chapter 4
The Selberg class
As indicated in the last but one section, Artin proved his conjecture if
χ is one-dimensional and L/K is abelian. In this case, the related Artin
L-function coincides with a Hecke L-function.
Theorem 4.17. Let L/K be abelian and let ρ 6= 1 be an irreducible character
of G = Gal(L/K). Then there exists a Hecke grössencharacter ψ such that
L(s, ρ, L/K) = L(s, ψ).
Artin proved this theorem by means of class field theory and, in particular,
Chebotarev’s density theorem. We shall briefly explain the latter result. Let
L/K be a finite Galois extension with Galois group G and let C be a subset
of G, closed under conjugation. Further, denote by πC (x) the number of
prime ideals p of K, unramified in L, for which σp ⊂ C and which have norm
N(p) ≤ x in K. Then, Chebotarev’s density theorem [31] states
♯C
π(x).
♯G
This rather deep theorem can be seen as a higher analogue of the prime
number theorem in arithmetic progressions. A modern proof can be found,
for example, in Narkiewicz [159]. The Chebotarev density theorem can be
used to determine the Galois group of a given irreducible polynomial P (X)
of degree n by counting the number of unramified primes up to a certain
bound for which P factors in a certain way and comparing the results with
the fractions of elements of each of the transitive subgroups of the symmetric
group Sn with the same cyclic structure; see Lenstra & Stevenhagen [131]
for details.
Brauer [25] proved a functional equation for Artin L-functions which gives
a meromorphic continuation throughout the complex plane (see also Neukirch
[160]), §VII.12). However, the holomorphy of nonabelian Artin L-functions
is still unproved (especially inside the critical strip). In certain particular
cases the Artin conjecture is known to be true, at least conditionally. M.R.
Murty [155] proved
(4.31)
πC (x) ∼
Theorem 4.18. Selberg’s Conjecture B implies Artin’s conjecture.
M.R. Murty & Perelli [158] replaced Selberg’s conjecture by the pair correlation conjecture (as already mentioned in the previous section).
The proof uses some easy properties of Artin L-functions which we did
not prove or even did not mention above. The reader may have a look into
the literature, e.g., Heilbronn [91], and may consult the examples from the
previous section.
Proof. Let L̃ be the normal closure of L over Q. Then, L̃/K and L̃/Q are
Galois. Thus, χ can be thought as a character χ̃ of Gal(L̃/K), and by the
Section 4.6
Artin L-functions
183
properties of Artin L-functions it turns out that
L(s, χ̃, L̃/K) = L(s, χ, L/K).
Brauer’s induction theorem [25] (see again Neukirch [160], §VII.10) states,
roughly, that any character χ̃ of a finite group G is a N0 -linear combination
of certain induced one-dimensional characters ψ of subgroups of G. Thus, by
the induction of χ̃ from Gal(L̃/K) to Gal(L̃/Q), it follows that
Y
L(s, χ, L/K) =
L(s, ψ, L̃/Q)m(ψ) ,
ψ
where the product is taken over all irreducible characters ψ of Gal(L̃/Q)
and m(ψ) are nonnegative integers. To prove Artin’s conjecture, it suffices
to show that all appearing L(s, ψ, L̃/Q) are entire. By Brauer’s induction
theorem and Artin’s reciprocity law, Theorem 4.17,
L(s, ψ, L̃/Q) =
L(s, χ1 )
,
L(s, χ2 )
where χ1 , χ2 are characters of Gal(L̃/Q) and L(s, χ1 ), L(s, χ2 ) are products
of Hecke L-functions (4.22). Since Hecke L-functions belong to the Selberg
class S, and S is multiplicatively closed, the functions L(s, χ1 ) and L(s, χ2 )
belong to S too. Now, by Theorem 4.12, there exist primitive functions
Lj ∈ S such that
(4.32)
L(s, ψ, L̃/Q) =
f
Y
j=1
Lj (s)ej ,
where ej ∈ Z. By comparing the p-th coefficient in the Dirichlet series expansions of both sides, we get
ψ(p) =
f
X
j=1
ej aLj (p).
Thus,
X |ψ(p)|2
(4.33)
p≤x
p
2
f
X 1 X
ej aLj (p) .
=
p j=1
p≤x
Selberg’s conjecture B yields the asymptotic formula
2
f
!
f
X 1 X
X
(4.34)
ej aLj (p) =
e2j log log x + O(1).
p
p≤x
j=1
j=1
Next, we decompose the sum on the left hand side of (4.33) according to
the conjugacy classes C of G := Gal(L̃/Q) to which the Frobenius element σp
Chapter 4
184
The Selberg class
belongs. If gC denotes any element of C, this leads to
X 1
X |ψ(p)|2 X
=
|ψ(gC)|2
.
p
p
p≤x
p≤x
C
σp ⊂C
By partial summation, we deduce from Chebotarev’s density theorem (4.31)
X 1
♯C
=
log log x + O(1).
p
♯G
p≤x
σp ⊂C
This gives
X |ψ(p)|2
p≤x
p
=
X
C
|ψ(gC)|2
♯C
log log x + O(1).
♯G
Since ψ is irreducible, we get
X
♯C
1 X
|ψ(gC )|2
♯C = 1,
=
♯G
♯G
C
C
which implies with (4.34) and (4.33)
f
X
e2j = 1.
j=1
Thus, f = 1 and e1 = ±1. The case e1 = −1 implies
1
,
L(s, ψ, L̃/Q) =
L1 (s)
which is impossible since L(s, ψ, L̃/Q) has trivial zeros (their existence follows
from their functional equation). Hence, e1 = +1, and we conclude that
L(s, ψ, L̃/Q) = L1 (s) is entire. •
The proof shows that if χ is an irreducible non-trivial character of Gal(K/Q),
then the Artin L-function L(s, χ, K/Q) is an element of the Selberg class S
if Selbergs’s Conjecture B is true. Moreover, one can easily show that under
these assumptions it is even primitive.
Exercise 111. Deduce from the discussion of the arithmetic of the Gaussian number field Q(i) an old statement of Fermat and Euler which states that an odd prime
p has a representation as a sum of two integer squares if and only if p ≡ 1 mod 4.
Exercise 112. Derive the functional equation for the Artin L-functions
1
2πi
L(s, χj , L/Q) for j = 1, 2, 3 in the example L = Q(2 3 , e 3 ) from Section 4.6.3.
Hint: use the functional equation for Dedekind zeta-functions.
√
√
√
Exercise 113. * Construct all Artin L-functions to Q( 10) and Q( 2 + 3).
Section 4.7
Langlands program
185
√
√
Exercise 114. Consider the field K := Q( −1, −5). Show that it has
√ three
√
√
different subfields of degree 2 over Q, namely Q( −1), Q( −5), and Q( 5), and
verify the identity
(4.35)
ζK (s)ζQ (s)2 = ζQ(√−1) (s)ζQ(√−5) (s)ζQ(√5) (s).
Prove that if the Galois group Gal(K/Q) has more normal subgroups than conjugacy
classes, then there exist algebraic relations for the corresponding Dedekind zetafunctions.
Exercise 115. Prove that Selberg’s Conjecture B implies that an Artin L-function
L(s, χ, K/Q) with an irreducible non-trivial character χ is primitive.
Hint: consider the corresponding integer nL in Conjecture A and recall an old
exercise.
4.7. Langlands program
The Langlands program has emerged in the late 60’s of the last century in a
series of far-reaching conjectures tying together seemingly unrelated objects
in number theory, algebraic geometry, and the theory of automorphic forms.
These disciplines are linked by Langlands’ L-functions associated with automorphic representations, and by the relations between the analytic properties
and the underlying algebraic structures. There are two kinds of L-functions:
motivic L-functions which generalize Artin L-functions and are defined purely
arithmetically, and automorphic L-functions, defined by transcendental data.
In its comprehensive form, an identity between a motivic L-function and an
automorphic L-function is called a reciprocity law. Langlands’ reciprocity
conjecture claims, roughly, that every L-function, motivic or automorphic, is
equal to a product of L-functions attached to automorphic representations.
For an introduction to the Langlands program we refer to the excellent surveys of Gelbart [58], M.R. Murty [154], and Langlands’ lecture [126] at the
International Congress in Helsinki.
4.7.1. Automorphic representations. At the heart of Langlands’s
program is the notion of an automorphic representation π and its L-function
L(s, π). It is beyond the scope of these notes to define these objects (both defined via group theory and the theory of harmonic analysis on adèle groups)
in an appropriate way.
Let K be a number field (one looses not too much by restricting to Q).
For each absolute value ν on K, there is a completion Kν of K which is R,
C, or a p-adic field, where p is a prime ideal in K. Denote by Oν the ring
of integers in Kν . In discussing local-global problems it is often necessary
to consider several places simultaneously. At first sight it seems natural to
Chapter 4
186
The Selberg class
form the product of all the Kν which is a topolgical ring, but it does not have
satisfactory compactness properties. Since any α ∈ K is a p-adic integer for
almost all p, we restrict to elements
Y
α=
αν ,
ν
where αν lies in Oν for all but finitely many places ν; such elements are called
adèle. The adèle form a set-theoretic (restricted) product. This product is a
topological ring, the adèle ring AK of K. One can think of K as embedded in
AK via the map α 7→ (α, α, . . .).
For m ≥ 1 let GLm (AK ) be the group of m × m matrices over AK whose
determinant is a unit in AK . By the product topology of the adèle ring,
GLm (AK ) becomes a locally compact group in which GLm (K), embedded diagonally, is a discrete subgroup of GLm (AK ). A character ψ of K∗ \ GL1 (AK )
is called grössencharacter, where K∗ := K \ {0}. For a fixed grössencharacter
ψ we consider the Hilbert space
L2 := L2 (GLm (K) \ GLm (AK ), ψ)
of measurable functions f on GLm (K) \ GLm (AK ) satisfying the conditions
• f (zg) = ψ(z)f (g) for any z ∈ Z, g ∈ GLm (K) \ GLm (AK );
• the integral
Z
ZGLm (K)\GLm (AK )
|f (g)|2 dg
is bounded.
Elements f ∈ L2 generalize the concept of twisted modular forms to discrete
subgroups of the full modular group. In order to introduce a subspace of cusp
forms we have to consider appropriate subgroups. Any parabolic subgroup
P of GLm (R), where R is a commutative ring with identity, has a decomposition, called the Levi decomposition, of the form P = MN, where N is the
unipotent radical of P ; M is called the Levi component of P . We denote the
unipotent radical of P in the Levi decomposition of a parabolic subgroup P
in GLm (R) by NP (R).
The subspace of cusp forms
L20 := L2 (GLm (K) \ GLm (AK ), ψ)
of L2 is defined by the additional vanishing condition
• for all parabolic subgroups P of GLm (AK ) and every g ∈ GLm (AK ),
Z
f (ng) dn = 0.
NP (K)\NP (AK )
The right regular representation R of GLm (K) on L2 is given by
(R(g)f )(α) = f (αg)
Section 4.7
Langlands program
187
for each f ∈ L2 and any α, g ∈ GLm (AK ). An automorphic representation
is a subquotient of the right regular representation of GLm (AK ) on L2 , and
a cuspidal automorphic representation is a subrepresentation of the right
regular representation of GLm (AK ) on L20 .
A representation of GLm (AK ) is called admissible if its restriction to the
maximal subgroup
Y
Y
Y
K :=
Um (C) ×
Om (R) ×
GLm (Oν )
ν complex
ν real
ν f inite
contains each irreducible representation of K with finite multiplicity; here
Um and Om denote the groups of unitary and orthogonal m × m matrices,
respectively. A representation π of a group G is called irreducible if it cannot
be decomposed into the direct sum of two representations; an irreducible
character is the character associated with an irreducible representation.
Now let π be an irreducible, admissible, cuspidal automorphic representation of GLm (K). Then π can be factored into a direct product π = ⊗ν πν ,
where ν ranges over all (finite and infinite) places of K, and each πν is an
irreducible representation of GLm (Kν ). For all but a finite number of places
ν the representation πν is unramified (that means the quotient obtained by
inducing a quasi-character from the Borel subgroup of GLm (Kν ) to GLm (Kν )
is unique).
4.7.2. General L-functions. In order to define the L-function attached
to an automorphic representation π we define the local Euler factors for nonarchimedean (finite) unramified places ν by
−1
Aν
,
Lν (s, π) = det 1 −
N(p)s
where Aν is the semisimple conjugacy class corresponding to πν and p is
the prime ideal of K belonging to the place ν. We do not explain here the
rather technical definition of the Euler factors Lν (s, π) for ramified places ν.
However, any Euler factor Lν (s, π) for a non-archimedean place ν associated
with the prime ideal p, unramified or not, can be rewritten as
−1
m Y
αj (p)
(4.36)
,
Lν (s, π) =
1−
N(p)s
j=1
where the numbers αj (p) for 1 ≤ j ≤ m are so-called Satake, resp. Langlands parameters, determined from the local representations πν . At the
archimedean (infinite) places ν we put for certain numbers αj (ν)
Lν (s, π) =
m
Y
j=1
Γν (s − αj (ν))
188
Chapter 4
The Selberg class
with
(4.37)
Γν (s) :=
s
if Kν ≃ R,
π − 2 Γ 2s
(2π)−s Γ(s) if Kν ≃ C;
where, again, the appearing numbers αj (ν) for 1 ≤ j ≤ m are determined
from the local representations πν . Then the global L-function associated with
π is given by
Y
Lν (s, π),
L(s, π) =
ν non−archimedean
and the completed L-function is defined by
Y
Λ(s, π) = L(s, π)
Lν (s, π).
ν archimedean
By the work of Hecke [88], Jacquet & Langlands [102], and Godement &
Jacquet [60] we have
Theorem 4.19. Let K be a number field and π be an irreducible, admissible, cuspidal automorphic representation of GLm (AK ). Then Λ(s, π) has
a meromorphic continuation throughout the complex plane and satisfies the
functional equation
s− 21
Λ(s, π) = ǫπ Nπ
Λ(1 − s, π̃),
where π̃ is the contragredient representation of π, Nπ ∈ N is the conductor of
π and ǫπ is the root number (these quantities are completely determined by
the local representations). Λ(s, π) is entire unless m = 1 and π is trivial, in
which case it has a pole at s = 1.
For m = 1 one simply obtains the Riemann zeta-function, Dirichlet Lfunctions and Hecke L-functions attached to grössencharacters, whereas for
m = 2 one gets L-functions associated with newforms. The similarities between these general L-functions and those of the Selberg class are obvious.
On one hand we have the Selberg class defined by axioms which are known
to be the most common pattern of many L-functions in number theory, on
the other hand we have Langlands’ construction of general L-functions out
of group representations.
4.7.3. Langlands’ conjectures. In the 1960’s Langlands started his visionary program which might be understood as a continuation of the famous
Artin conjecture. One of his central conjectures claims that all zeta-functions
arising in number theory are special realizations of L-functions to automorphic representations constructed above.
Langlands’ reciprocity conjecture. Suppose L is a finite Galois extension
of a number field K with Galois group G, and ρ : G → V is an irreducible
Section 4.7
Langlands program
189
representation of G, where V is an m-dimensional vector space. Then there
exists an automorphic cuspidal representation π of GLm (AK ) such that
L(s, ρ, L/K) = L(s, π).
This means that there are identies between certain L-functions, which are a
priori of different type! Since Hecke grössencharacters are automorphic representations of GL1 (A), Artin’s conjecture is a special case of the Langlands
reciprocity conjecture. By Artin’s work, if m = 1 and L/K is abelian, Langlands’ reciprocity law is settled by means of class field theory. In the case of
function fields, the Langlands conjecture has been proved by Drinfeld [48] in
dimension two, and recently by Lafforgue [123] for arbitrary dimension (for
which both of them were awarded with a Fields medal).
Now we consider the local Euler factors of L-functions attached to automorphic representations. Petersson [169] extended Ramanujan’s conjecture
on the values of the τ -function to modular forms. Deligne’s estimate (3.32)
proved the desired bound for newforms but it is expected that an analogue
should hold for all L-functions of arithmetical nature.
Ramanujan-Petersson Conjecture. Let π be a cuspidal automorphic
representation of GLm (AK ) which is unramified at a place ν. If ν is nonarchimedean, then
|αj (p)| = 1
for
1 ≤ j ≤ m,
where p is the prime ideal associated with the place ν. If ν is archimedean,
then Re αj (ν) = 0 for 1 ≤ j ≤ m.
The Ramanujan-Petersson conjecture might look very restrictive on the first
view, but it is nothing else than the local analogue of the Grand Riemann
hypothesis. We refer to Iwaniec & Sarnak [100] for details and the current
knowledge concerning this conjecture.
We shall speculate a little bit about all these widely believed conjectures
and the axioms defining the Selberg class and the subclass S̃, in particular. It is expected that all functions in the Selberg class are automorphic
L-functions. If L ∈ S is primitive and automorphic, then it is also attached
to an irreducible automorphic representation. Conversely, every irreducible
automorphic representation should give a primitive function in S. This is
not known in general, but it has been proved by M.R. Murty [155, 156]
for GL1 and GL2 . The axioms on the analytic continuation and on the functional equation follow immediately from Theorem 4.19. The polynomial Euler
product in the definition of the subclass S˜ fits (by the splitting of primes in
K) perfectly to the Euler product of Langlands’ L-functions attached to automorphic representations (4.36) and the Ramanujan-Petersson conjecture.
190
Chapter 4
The Selberg class
Finally, let us notice that the Euler factor at the infinite places (4.37) is of
the form, predicted by the strong λ-conjecture (from Section 4.2). Of course,
all these axioms and the hypotheses too, are deduced from known examples
of L-functions in number theory, and so they have to share certain patterns.
Anyway, we are led to see a close connection between Langlands’ general
L-functions and the elements of the Selberg class.
M.R. Murty [155] proved
Theorem 4.20. Assume that Selberg’s Conjecture B is true.
i) If π is an irreducible cuspidal automorphic representation of GLm (AQ )
which satisfies the Ramanujan-Petersson conjecture, then L(s, π) is a primitive function in S.
ii) If K is a Galois extension of Q with solvable Galois group G, and if χ
is an irreducible character of G of degree m, then there exists an irreducible
cuspidal automorphic representation π of GLm (AQ ) such that
L(s, χ) = L(s, π).
The first assertion identifies certain L-functions to automorphic representations as being primitive functions in the Selberg class subject to the truth of
Selberg’s conjecture B and the Ramanujan-Petersson conjecture. The second assertion of the theorem is Langland’s reciprocity conjecture if K/Q is
solvable. Murty’s proof shows that if the Dedekind zeta-function of K is the
L-function of an automorphic representation over Q, then Selberg’s Conjecture B implies Langlands’ reciprocity conjecture.
Some concluding words: We started with factorizations of integers (first in Q
and then in number fields) into irreducible elements and derived asymptotic
laws for the atoms in these products (that were primes or prime ideals) by
studying the analytic properties of the generating functions (Euler products).
Finally, we considered factorizations of more complicated higher L-functions
(Artin – vs. Hecke L-functions) into primitive functions and deduced analytic
properties (which are not unrelated to number theory; however this came a
bit short here) by applying arithmetical laws. A fruitful see-saw!
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