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Outline
Prime Numbers
Riemann zeta-function
Reasons for belief and disbelief
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Riemann’s Memoir and the Related
Liangyi Zhao
Division of Mathematical Sciences
School of Physical and Mathematical Sciences
Nanyang Technological University
Singapore
18 November 2009
Liangyi Zhao
Riemann’s Memoir and the Related
Outline
Prime Numbers
Riemann zeta-function
Reasons for belief and disbelief
References
1
Prime Numbers
Infinitude by Euclid
Infinitude by Euler
Prime Number Theorem
2
Riemann zeta-function
Definition
Riemann’s Memoir
The Riemann Hypothesis
The Explicit Formula
Proof of PNT
3
Reasons for belief and disbelief
Reasons for Belief
The Devil’s Advocate
Lehmer’s Phenonemon
4
References
Liangyi Zhao
Riemann’s Memoir and the Related
Outline
Prime Numbers
Riemann zeta-function
Reasons for belief and disbelief
References
Infinitude by Euclid
Infinitude by Euler
Prime Number Theorem
Infinitude by Euclid of Alexandria circa 22 B. C.
A prime number is a natural number greater than one that is
divisible only by one and itself. Let P henceforth denote the
set of prime numbers.
The fundamental theorem of arithmetics (FTA) states that
every natural number can be uniquely, up to re-ordering the
factors, written as a product of primes.
Consequently, since n! + 1 has no divisor between 2 and n
there exists a prime (dividing n! + 1) greater than n.
Therefore, there must be infinitely many prime numbers.
Liangyi Zhao
Riemann’s Memoir and the Related
Outline
Prime Numbers
Riemann zeta-function
Reasons for belief and disbelief
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Infinitude by Euclid
Infinitude by Euler
Prime Number Theorem
Infinitude by Euler in 1737
For n > 1, we have
Y
Z n
n
X
Y
1
1
1
1
1
dx ≤
≤
1 + + 2 + ··· ≤
log n =
.
1
k
p p
1
−
1 x
p
p≤n
p≤n
k=1
p∈P
p∈P
Upon taking the logarithm of both side of the above, we have
log log n ≤ −
X
log(1 − p
−1
p≤n
p∈P
∞
XX
1
,
)=
lp l
p≤n l=1
p∈P
where we have used the Taylor series
log(1 − x) = −
∞
X
xl
l=1
Liangyi Zhao
l
, for |x| < 1.
Riemann’s Memoir and the Related
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Prime Numbers
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Infinitude by Euclid
Infinitude by Euler
Prime Number Theorem
Infinitude by Euler in 1737
Now we have
log log n ≤
∞
∞
XX
X 1 XX
1
1
=
+
.
l
p
lp
lp l
p≤n l=1
p∈P
p≤n
p∈P
p≤n l=2
p∈P
We can estimate the last sum above in the following way.
n X
∞
n
∞
∞
XX
X
X
X
1
1
1
−l
≤
m
=
≤
.
2
−1
l
m (1 − m )
m(m − 1)
lp
p≤n l=2
p∈P
m=2 l=2
m=2
m=2
Note that the last sum converges and the limit is 1.
Liangyi Zhao
Riemann’s Memoir and the Related
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Infinitude by Euclid
Infinitude by Euler
Prime Number Theorem
Infinitude by Euler in 1737
Therefore, we must have,
log log n ≤
X1
+ θ, with |θ| ≤ 1.
p
p≤n
p∈P
This means that as n tends to infinity,
X1
→ ∞.
p
p≤n
p∈P
Hence there must be infinitely many prime numbers as finite
sums converge.
Liangyi Zhao
Riemann’s Memoir and the Related
Outline
Prime Numbers
Riemann zeta-function
Reasons for belief and disbelief
References
Infinitude by Euclid
Infinitude by Euler
Prime Number Theorem
How infinite are the prime numbers?
Let π(x) denote the number of primes not exceeding x.
Modifying Euclid’s proof would give π(x) ≥ log log x.
Euler’s proof would suggest that π(x) should be much larger.
It was first conjectured by Legendre that the ratio of π(x) and
x
x
log x is 1 as x → ∞. Hence π(x) is well-approximated by log x
if x is large.
Gauss wrote in 1849 that he reached the conclusion of the
conjecture at the age of 15 in 1792, although what he
believed was that π(x) is well-approximated by
Z x
1
dt.
li(x) =
2 log t
Liangyi Zhao
Riemann’s Memoir and the Related
Outline
Prime Numbers
Riemann zeta-function
Reasons for belief and disbelief
References
Infinitude by Euclid
Infinitude by Euler
Prime Number Theorem
How infinite are the prime numbers?
It can be shown, using integration by parts, that
x
x
+
[1 + o(1)], as x → ∞.
li(x) =
log x
log2 x
It was in 1896 that Hadamard and de la Vallée Poussin proved
the conjecture independently.
Theorem (Prime Number Theorem)
There exists a constant c > 0, effectively computable such that for
x ≥2
p
π(x) = li(x) + O x exp −c log x ,
where the implied constant is absolute.
Liangyi Zhao
Riemann’s Memoir and the Related
Outline
Prime Numbers
Riemann zeta-function
Reasons for belief and disbelief
References
Infinitude by Euclid
Infinitude by Euler
Prime Number Theorem
How good is this approximation?
x
108
109
1010
1011
1012
1013
1014
1015
1016
π(x)
5,761,455
50,847,534
455,052,511
4,118,054,813
37,607,912,018
346,065,536,839
3,204,941,750,802
29,844,570,422,669
279,238,341,033,925
Liangyi Zhao
[li(x) − π(x)]
754
1,701
3,104
11,588
38,263
108,971
314,890
1,052,619
3,214,632
Riemann’s Memoir and the Related
Outline
Prime Numbers
Riemann zeta-function
Reasons for belief and disbelief
References
Definition
Riemann’s Memoir
The Riemann Hypothesis
The Explicit Formula
Proof of PNT
Riemann zeta-function
Let
ζ(s) =
∞
X
Y
−1
1
=
1 − p −s
, for<s > 1.
s
n
n=1
p∈P
The infinite sum and product are the same by FTA.
It is much more natural to count the primes with a weight.
Set
log p; if n = p k ,
Λ(n) =
0,
otherwise.
The prime number theorem is equivalent to
p
X
ψ(x) :=
Λ(n) = x + O x exp −c log x .
n≤x
Liangyi Zhao
Riemann’s Memoir and the Related
Outline
Prime Numbers
Riemann zeta-function
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Definition
Riemann’s Memoir
The Riemann Hypothesis
The Explicit Formula
Proof of PNT
Riemann zeta-function
The reason for counting primes this way is that it brings ζ(s)
into the picture.
−
X
ζ 0 (s)
d
p −s
= − log ζ(s) =
log p
ζ(s)
ds
1 − p −s
p∈P
=
X
log p p
−s
+p
−2s
+ ··· =
∞
X
Λ(n)
n=1
p∈P
ns
, for <s > 1.
Euler was the first to study ζ(s), only considering the function
as that of a real variable.
Riemann introduced complex analysis to the investigation and,
in doing so, set a new direction for mathematics.
Liangyi Zhao
Riemann’s Memoir and the Related
Outline
Prime Numbers
Riemann zeta-function
Reasons for belief and disbelief
References
Definition
Riemann’s Memoir
The Riemann Hypothesis
The Explicit Formula
Proof of PNT
Riemann’s Memoir
In 1859, Georg Friedrich Bernhard Riemann, a newly elected
member of the Berlin Academy of Sciences reporting on his
most recent research, sent an article titled Ueber die Anzahl
der Primzahlen unter einer gegebenen Grösse (On the Number
of Primes Less than a Given Magnitude) to the academy.
Considering that it was his only paper in the theory of
numbers and changed the direction of mathematical research
in very significant ways, it is now appropriately and better
known as “Riemann’s Memoir.”
The reason for this dubbing is perhaps also partly due to the
fact that Riemann fell seriously ill two years later and passed
on in 1866, almost two months before his fortieth birthday.
Liangyi Zhao
Riemann’s Memoir and the Related
Outline
Prime Numbers
Riemann zeta-function
Reasons for belief and disbelief
References
Definition
Riemann’s Memoir
The Riemann Hypothesis
The Explicit Formula
Proof of PNT
Riemann’s Theorems
ζ(s) can be continued meromorphically to the whole of C
with only one pole at s = 1 of residue 1.
Z ∞
s
ζ(s) =
−s
{x}x −s−1 dx, for <s > 0.
s −1
1
ζ(s) satisfies the functional equation
s 1−s
− 2s
− 1−s
ζ(s) = π 2 Γ
π Γ
ζ(1 − s).
2
2
Hence the values of ζ(s) for <s < 0 can be obtained from
those for <s > 1. In particular ζ(s) = 0 if s is a negative even
integer since Γ(s/2) has a pole there. These are known as the
trivial zeros.
Liangyi Zhao
Riemann’s Memoir and the Related
Outline
Prime Numbers
Riemann zeta-function
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Definition
Riemann’s Memoir
The Riemann Hypothesis
The Explicit Formula
Proof of PNT
Riemann’s Conjectures
ζ(s) has infinitely many zeros ρ with 0 < <ρ < 1 (the critical
strip), symmetric about the lines =s = 0 and <s = 21 . The
symmetry follows from Riemann’s theorems. The infinitude is
a consequence of the next conjecture.
Let N(T ) be the number of zeros ρ of ζ(s) with 0 ≤ <ρ ≤ 1
and 0 ≤ =ρ ≤ T . Then
N(T ) =
T
T
T
log
−
+ O(log T ).
2π
2π 2π
This is known as the Riemann-von Mangoldt formula which
was first proved in 1895 by von Mangoldt in a less satisfying
form and fully in 1905.
Liangyi Zhao
Riemann’s Memoir and the Related
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Definition
Riemann’s Memoir
The Riemann Hypothesis
The Explicit Formula
Proof of PNT
Riemann’s Conjectures in his Memoir
The function
s s
1
ξ(s) = s(s − 1)π − 2 Γ
ζ(s)
2
2
is entire and satisfies a product formula
Y
s
s
ξ(s) = exp(A + Bs)
1−
exp
,
ρ
ρ
ρ
where A, B ∈ R and ρ runs over the zeros of ζ(s) in the
critical strip. This was proved by Hadamard in 1893.
There is an explicit formula for π(x) − li(x), or equivalently
X x ρ ζ 0 (0) 1
1
ψ(x) − x = −
−
− log 1 − 2
ρ
ζ(0)
2
x
ρ
which was proved in 1895 by von Mangoldt.
Liangyi Zhao
Riemann’s Memoir and the Related
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Definition
Riemann’s Memoir
The Riemann Hypothesis
The Explicit Formula
Proof of PNT
The Riemann hypothesis
The only unsolved conjecture in Riemann’s Memoir is the
Riemann hypothesis(RH). It states that all the non-trivial
zeros of ζ(s) have real part 21 .
It is not clear what led Riemann to this conjecture, or any of
the ones mentioned above. But it seems that Riemann knew a
lot more about ζ(s) than is apparent in the published memoir.
Riemann was cautious in his memoir, using the words ”very
likely”(”sehr wahrscheinlich”) in connection with RH.
”One would of course like to have a rigorous proof of this
[RH], but I have put aside the search for such a proof after
some fleeting vain attempts because it is not necessary for the
immediate objective of my investigation.”
Liangyi Zhao
Riemann’s Memoir and the Related
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Definition
Riemann’s Memoir
The Riemann Hypothesis
The Explicit Formula
Proof of PNT
The Riemann’s hypothesis
Hardy in 1914 showed that infinitely many zeros lie on the
critical line, the line with <s = 12 .
A more precise statement of this was later given by Hardy and
Littlewood, that the number of zeros on the critical line with
imaginary parts in [0, T ] is greater than cT for some c > 0.
In 1942, A. Selberg proved that a positive proportion of
non-trivial zeros have real part 21 .
In 1974, Levinson showed that at least one third of all zeros
are at the correct place.
Conrey improved this proportion to 40% in 1991.
No counter example to RH has been found in the first
1.5 × 109 zeros of ζ(s) in the critical strip.
Liangyi Zhao
Riemann’s Memoir and the Related
Outline
Prime Numbers
Riemann zeta-function
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Definition
Riemann’s Memoir
The Riemann Hypothesis
The Explicit Formula
Proof of PNT
Proof of the Explicit Formula
If c > 0, then.

Z
Z
 0, if 0 < y < 1,
1
s ds
1/2,
if y = 1,
y
=
where
=
.

2πi (c)
s
(c)
<s=c
1,
if y > 1,
P
Therefore, if c > 1, ψ(x) = n≤x Λ(n)
Z Z
∞
x s ds
−1
ζ 0 (s) x s
1 X
Λ(n)
=
ds.
≈
2πi
s
2πi (c) ζ(s) s
(c) n
Z
n=1
Move the contour to the left and collect all residues, we get
X x ρ ζ 0 (0) 1
1
ψ(x) − x = −
−
− log 1 − 2 .
ρ
ζ(0)
2
x
ρ
Liangyi Zhao
Riemann’s Memoir and the Related
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Riemann zeta-function
Reasons for belief and disbelief
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Definition
Riemann’s Memoir
The Riemann Hypothesis
The Explicit Formula
Proof of PNT
Proof of the Explicit Formula
From the explicit formula below, it is important, in order to
know the size of the sum over ρ; we need an upper bound for
the real parts of these ρ’s.
X x ρ ζ 0 (0) 1
1
ψ(x) − x = −
−
− log 1 − 2
ρ
ζ(0)
2
x
ρ
The best case scenario, because of the symmetry of the ρ’s, is
<ρ = 1/2 for all ρ.
Liangyi Zhao
Riemann’s Memoir and the Related
Outline
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Definition
Riemann’s Memoir
The Riemann Hypothesis
The Explicit Formula
Proof of PNT
Proof of PNT
We use a truncated version of the discrete integral.
This gives
X xρ
ψ(x) − x =
+ error terms.
ρ
|=ρ|<T
A zero-free region enables us to get the error term in PNT.
There is c > 0 such that if ρ = β + iγ with β > 0 and
ζ(ρ) = 0, then β < 1 − c/ log γ.
Partial summation allows us to go from ψ(x) to π(x).
RH would imply the error term in PNT can be taken to be
√
O
x log x .
Liangyi Zhao
Riemann’s Memoir and the Related
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Reasons for Belief
The Devil’s Advocate
Lehmer’s Phenonemon
Reasons for Belief
A sanguine disposition: mathematics should be beautiful, the
distribution of primes should be as ”regular” as possible.
One can infer
Euler product that for <s > 1,
P from that
−s where µ(n) is zero if n is not
ζ −1 (s) = ∞
µ(n)n
n=1
square-free and (−1)k if n is the product of k distinct primes.
RH is equivalent to
X
µ(n) = Oε x 1/2+ε .
n≤x
µ(n) appears to take on ±1 fairly randomly and hence one
may expect the sum above to have a lot of cancellations,
giving the desired estimate.
Liangyi Zhao
Riemann’s Memoir and the Related
Outline
Prime Numbers
Riemann zeta-function
Reasons for belief and disbelief
References
Reasons for Belief
The Devil’s Advocate
Lehmer’s Phenonemon
Reasons for Belief
A VERY large number of zeros have been computed and no
counter example to RH has been found.
A large proportion can be proved to be very close to the line,
i. e. zero density theorems.
RH-like statements are known to hold for other functions
similar to ζ(s), e. g. zeta functions for function fields (Weil),
algebraic varieties over finite fields (Deligne).
Certain average versions of RH have been proved, e. g.
Bombieri-Vinogradov theorem,
Liangyi Zhao
Riemann’s Memoir and the Related
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Reasons for Belief
The Devil’s Advocate
Lehmer’s Phenonemon
Playing the Devil’s Advocate
RH-like statements are known NOT to hold for some
functions with properties similar to those of ζ(s), e.g.
Davenport-Heilbronn zeta-function, Epstein zeta-function.
The hitherto also unresolved Lindelöf hypothesis (LH), which
is a consequence of RH, asserts
ζ(1/2 + it) = Oε (t ε ) for any ε > 0.
There are heuristics that would lead to the falsehood LH.
True behaviors of ζ(s) may not be apparent for s with small
imaginary part. RH implies that the error term in the
Riemann-von Mangoldt formula may be taken to be
O (log x/ log log x), but the actual error (known to be
unbounded) has never been seen to be much larger than 3.
Liangyi Zhao
Riemann’s Memoir and the Related
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Prime Numbers
Riemann zeta-function
Reasons for belief and disbelief
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Reasons for Belief
The Devil’s Advocate
Lehmer’s Phenonemon
Lehmer’s Phenomenon
We consider a certain real-valued Z (t) with
|Z (t)| = |ζ(1/2 + it)|.
RH implies that Z (t) has no positive local minimum or
negative local maximum for t > 1000.
The term ”Lehmer’s phenomenon” refers to a behavior of
Z (t) that its graph sometimes barely crosses the t-axis, an
almost counter example of RH.
This shows the delicacy of ζ(s) and ”must give pause to even
the most convinced believer in the Riemann hypothesis.”
Liangyi Zhao
Riemann’s Memoir and the Related
Outline
Prime Numbers
Riemann zeta-function
Reasons for belief and disbelief
References
Further Readings
H. M. Edwards, Riemann’s Zeta Function
A. Ivić, The Riemann Zeta-function: Theory and Applications
S. J. Patterson, An Introduction to the Theory of the
Riemann Zeta-function.
E. C. Titchmarsh, The Theory of the Riemann Zeta-function
Liangyi Zhao
Riemann’s Memoir and the Related