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The Riemann Hypothesis
by Pedro Pablo Pérez Velasco, 2011
Introduction
Consider the series of questions:
● How many prime numbers (2, 3, 5, 7, ...) are there
less than 100, less than 10.000, less than 1.000.000?
● How many less than any given number X?
According to Zagier, astonishingly:
● Randomness: Where will the next prime sprout?
● Regularity: Precise laws govern their behaviour.
Primes are, multiplicatively speaking, the building
blocks from which all numbers can be made. Reason:
Any number can be uniquely factored as a product of
primes (except for the order).
Primes relate additive and multiplicative structures.
Introduction
Simple unknown questions about primes:
Are there infinitely many pairs of primes whose
difference is 2? or 4? or 2k?
●
Is every even number greater than 2 a sum of two
primes? (Goldbach's conjecture)
●
Are there infinitely many primes which are 1 more
than a perfect square?
●
●
Is there some neat formula giving the next prime?
Sieves
Techniques to estimate the size of sets of integers
●
Erathostenes (remove multiples of primes),
Atkin,
Selberg,
Turán,
Sundaram,
Legendre,
Brun...
Objective: Find a smooth curve that is a reasonable
approximation to the staircase of primes
●
 x=number of primes less than x
First Approach to  x
Probabilistic approach due to Gauss
G(x) is proportional to x divided by the number of
digits of x, i.e.
●
x
 x≈
log  x 
x
 x
G x −G
1e02 25.75
25
0.75
1e03 166.5
167
-0.5
1e06 78626
78498
126
3e06 216745 216970
-135
The Riemann Hypothesis
●
●
First formulation
If
x
Li  x =∫2
x
1 /2
 x=
O x log  x
log  x
1
dt , a second formulation would be
log t
 x=Li  xO x
Notice that

x
Li  x =O
log  x

1/ 2
log  x
The Prime Number Theorem
●
It is the asymptotic law of distribution of primes

x
x
 x=
o
log  x
log  x 

or equivalently  x=Li  xo x/log  x 
●
Previous work by Legendre, Gauss, Chebyshev,...
The theorem was first proved by Hadamard and de la
Vallée Poussin in 1896, extending ideas of Riemann.
●
Many different proofs currently known. Erdös and
Selberg gave an elementary proof (1949, though long)
and Newman a short proof (1980, not elementary).
●
Fourier Analysis
●
Consider the trigonometric function
f t=a cos bt
where a is the amplitude and  is the frequency.
Fourier Analysis
Using linear combinations of basic trigonometric
functions we can approximate/define general functions
●
∞
f t=∑ a n cos n t
n=0
One interesting point is that by keeping track of
the amplitudes and the spectrum (frequencies) we can
encode most „reasonable“ functions => compression
●
The operation that starts with a graph and goes to
its spectral picture is the Fourier transform.
●
Fourier Analysis
How much „ cos t  “ occurs in f(t)? This is the
amplitude (Fourier transform) f  and is given by:
●
∞
f = ∫ f t  cos− t dt
−∞
The spectrum of f(t) is the set of frequencies
where the amplitude is nonzero.
●

The inverse operation (inverse Fourier transform) is
given by
●
∞
f t= ∫ f  cos t d 
−∞
Distributions
An integrable function is one which has values and
areas under its graph. A distribution D(t) has areas
under its graph (previous first condition relaxed) so
it makes sense to write b
●
∫ Dt 
a
The space of distributions was formaly defined and
first studied by Laurent Schwartz. The most prominent
distribution is the Dirac  function (which is not a
function!)
●
Distributions
The Dirac distribution enjoys many properties:
It is the derivative of the
Heaviside function.
●
●
02 1
Kinda „integral“ evaluation:
∞
f  x= ∫ f t  x t  dt
−∞
NB: The delta function can be rigurously defined as
a measure, not absolutely continuous wrt the Lebesgue
measure (previous formula is an abuse of notation).
●
Its derivative can be easily calculated using
'
integration by parts: 0 []=0 [' ]= ' 0.
●
●
Distributions are closed under derivations.
Distributions
●
Its Fourier transform is 1, and it is self-adjoint:
∞
= ∫  x e−2  i x dx=1
 x =1
〈  ,  〉 =〈  ,  〉
−∞
The delta function is an example of white noise: Every frequency
occurs in its Fourier series and they al occur in equal amounts
The spike function is d x  t= x t x −t / 2 with
Fourier transform:
●
d x =cos  x 
From the Primes to the Spectrum
All the information of the primes
staircase is not in the jumps but
in where those jumps happen.
●
Consider the function  x with
jumps of height log(p) for every x
which is a power of p.
●
Another formulation of the Riemann
hypothesis would be:
●
1 /2
 x=xO  x
●
2
log  x 
We further distort this function:
x
 x= e 
From the Primes to the Spectrum
The process ends by symmetrizing, rescaling the
function (multiplication by negative exponential) and
introducing Dirac deltas in the powers of primes:
●
−t/ 2
 t=e
d t
dt
From the Primes to the Spectrum
●
Let q = p n be such that q is less than C, then
C t =2 ∑ p−n /2 log  p d n log p  t
spike function
q
We are just considering the first C linear
combinations of d x  t functions. Its Fourier
transform is
●
C t =2 ∑ p−n /2 log  p cos n log  p 
q
The coordinates seem to be clustered about a
discrete set of positive real numbers  i . They are
known as the spectrum of the primes.
●
From the Primes to the Spectrum
From the Spectrum to the Primes
To recover the primes out of the spectrum, consider
the Fourier series with spectrum  i represented in a
logarithmic scale:
●
H C s =1 ∑ cosn · log s
nC
Riemann's way to build  x
 x solely from {n}n ∈ℕ?
●
Can we construct
●
Riemann's guess for an approximation to
 x is
∞
n
1 /n
R x=∑
Li  x 
n
n=1
where n is the Möbius function, with value 1 if n
has an even number of distinct prime factors, -1 if
the number of distinct primes is odd and 0 if not
square-free.
Riemann's way to build  x
Riemann's R(x) is a (conjecturally) much better
approximation than Li(x) or G(x).
●
Think of R(x) as the fundamental approximation to  x .
●
Riemann gave an infinite
sequence of guesses, all of
them are (conjecturally)
square root approximations:
●
R j  x=R j−1  x −R x
∞
 x=R x −∑ R x
j=1
1
i  j
2

1
i  j
2

exact formula
Riemann's way to build  x
Riemann's way to build  x
Riemann's way to build  x
Gracias por la atención
Prácticamente todo el material se
ha cogido (sin permiso) del libro
de B. Mazur y W. Stein
What is Riemann's Hypothesis?
disponible en la url
http://wstein.org/rh/rh/rh.pdf