Outline Introduction Crash Course in Number Theory Zeta Function Roots of the Zeta Function Prime Number Theorem Riemann Hy
Zeta Function
and Riemann Hypothesis
Ivan G. Avramidi
Department of Mathematics
New Mexico Tech
October 25, 2013
Outline Introduction Crash Course in Number Theory Zeta Function Roots of the Zeta Function Prime Number Theorem Riemann Hy
1
Introduction
2
Crash Course in Number Theory
3
Zeta Function
4
Roots of the Zeta Function
5
Prime Number Theorem
6
Riemann Hypothesis
7
Randomness in Prime Numbers
8
Random Hermitian Matrices
9
Free Riemann Gas
Outline Introduction Crash Course in Number Theory Zeta Function Roots of the Zeta Function Prime Number Theorem Riemann Hy
History of Prime Numbers
1
Euclid (300 BC)
2
Fermat (1640)
3
Euler (1737)
4
Dirichlet (1837)
5
Gauss (1849)
6
Chebyshev (1848)
7
Riemann (1859)
8
Dr X (2020)?
Outline Introduction Crash Course in Number Theory Zeta Function Roots of the Zeta Function Prime Number Theorem Riemann Hy
Millenium Prize Problem
B. Riemann, “On the number of primes less that a given
magnitude” (1859), 8 pages
One of twenty-three most important mathematical problems
composed by David Hilbert in 1900.
One of the seven Millennium Prize Problems of the Clay
Mathematics Institute (1 Million Dollar prize!) composed in
2000.
Outline Introduction Crash Course in Number Theory Zeta Function Roots of the Zeta Function Prime Number Theorem Riemann Hy
Evidence for the Riemann Hypothesis
Strong numerical evidence. Riemann hypothesis holds for
over 108 zeros at heights up to 1020 .
The exceptions to the Riemann hypothesis, if they exist, must
be rare.
More than 40% of the nontrivial zeros are simple and satisfy
the Riemann hypothesis.
Profoundly deep relation to such diverse areas of
mathematics as: analytic number theory, fractal geometry,
noncommutative geometry, random matrix theory, chaotic
dynamical systems (classical and quantum), quantum theory,
solid state physics, quasi-crystals, etc
Outline Introduction Crash Course in Number Theory Zeta Function Roots of the Zeta Function Prime Number Theorem Riemann Hy
Prime Numbers
A positive integer d divides a positive integer n, denoted by
d |n, if there is an integer k such that n = kd.
A positive integer d is the greatest common divisor of
positive integers a and b, denoted by d = (a , b ) if it is a
common divisor of a and b and is divisible by any other
common divisor.
Two positive integers a and b are relatively prime if
(a , b ) = 1.
A positive integer p > 1 is prime if it is divisible only by 1 and
p.
Outline Introduction Crash Course in Number Theory Zeta Function Roots of the Zeta Function Prime Number Theorem Riemann Hy
Notation
Summation over all positive divisors of n is denoted by
X
f (d )
d |n
Summation over all positive integers relatively prime to n is
denoted by
X
f (k )
( k ,n )
Summation over all primes is denoted by
X
f (p )
p
Summation over all powers of primes is denoted by
X
pk
f (n)
Outline Introduction Crash Course in Number Theory Zeta Function Roots of the Zeta Function Prime Number Theorem Riemann Hy
Fundamental Theorem of Arithmetic
The primes form an increasing sequence (pj )∞
j =1
p1 < p2 < p3 < · · ·
The study of this sequence as j → ∞ is central problem of
number theory.
Unique Factorization Theorem. Every positive integer n > 1
can be identified with a sequence of non-negative integers
(nj )∞
, nj ≥ 0, containing only finitely many non-zero terms
j =1
such that
n=
∞
Y
j =1
n
pj j
Outline Introduction Crash Course in Number Theory Zeta Function Roots of the Zeta Function Prime Number Theorem Riemann Hy
Examples
If all nj = 0, then n = 1
If only one nj is nonzero and nj = 1, then n = pj is a prime
If only one nj is non-zero and nj = k > 1, then n = pjk is a
power of prime
If all non-zero nj are equal to 1, then n is the product of
distinct primes
Outline Introduction Crash Course in Number Theory Zeta Function Roots of the Zeta Function Prime Number Theorem Riemann Hy
Arithmetic Functions
Arithmetic function is a real valued function
f : Z+ → R
defined on the set of positive integers that expresses some
arithmetical property of integers.
All arithmetic functions can be extended to functions of
positive real numbers, f : R → R as step functions by
f (x ) = f ([x ])
if x is not an integer and equal to the average of the left and
right values at the jumps.
Outline Introduction Crash Course in Number Theory Zeta Function Roots of the Zeta Function Prime Number Theorem Riemann Hy
Examples of Arithmetic Functions
The unit function
e (1) = 1,
and
e (n ) = 0
for n > 1
Function ω(n) is defined as the number of distinct prime
divisors of n.
Möbius function
µ(1) = 1,
and
µ(n) = (−1)ω(n)
if all primes divisors of n > 1 are distinct and zero otherwise.
The Von Mangoldt function
Λ(n) = log p
if n = p k for some prime p with some k ≥ 1 and zero
otherwise.
Outline Introduction Crash Course in Number Theory Zeta Function Roots of the Zeta Function Prime Number Theorem Riemann Hy
Dirichlet Convolution
The Dirichlet convolution h = f ∗ g of two arithmetic
functions, f , g, is defined by
h (n) =
X
f (d )g
d |n
n
d
The convolution inverse of an arithmetic function f is an
arithmetic function g such that its convolution with f is equal to
the unit function,
f ∗g =e,
that is,
X
d |n
f (d )g
n
d
= e (n)
In particular, g (1) = 1/f (1). So, if f (1) = 0, then it does not
exist.
Outline Introduction Crash Course in Number Theory Zeta Function Roots of the Zeta Function Prime Number Theorem Riemann Hy
Möbius Inversion
Fundamental Property of the Möbius Function
The Möbius function is the convolution inverse of the constant
function 1, that is, µ ∗ 1 = e or
X
µ(d ) = e (n)
d |n
It allows us to invert relations between arithmetic functions
n
X
X
F (n ) =
f (d )
iff
f (n) =
µ(d )F
d
d |n
d |n
One can also show that
g (x ) =
∞
X
n=1
f (nx )
iff
f (x ) =
∞
X
n =1
µ(n)g (nx )
Outline Introduction Crash Course in Number Theory Zeta Function Roots of the Zeta Function Prime Number Theorem Riemann Hy
Summatory Functions
Given an arithmetic function f one defines a function of a real
variable x ≥ 0,
X
Mf ( x ) =
f (n),
n ≤x
One is interested in the asymptotic behavior as x → ∞
Outline Introduction Crash Course in Number Theory Zeta Function Roots of the Zeta Function Prime Number Theorem Riemann Hy
Examples
The Mertens function
M (x ) =
X
µ(n)
n ≤x
The prime counting function
π(x ) =
X
1.
p ≤x
The weighted prime counting function
Π(x ) =
X 1
p k ≤x
k
=
X Λ(n)
.
log n
n ≤x
The Chebyshev function
ψ(x ) =
X
p k ≤x
log p =
X
n ≤x
Λ(n) .
Outline Introduction Crash Course in Number Theory Zeta Function Roots of the Zeta Function Prime Number Theorem Riemann Hy
Dirichlet Series
The asymptotics of a summatory function Mf of an arithmetic
function f as x → ∞ are described by the analytical properties
of the function of a complex variable s = σ + it,
Df (s ) =
∞
X
f (n )
ns
n=1
,
σ>c
called the Dirichlet Series of f .
The Dirichlet series Df (s ) and the summatory function Mf (x )
are related by the Mellin transform
∞
Z
dx x −s −1 Mf (x ),
Df (s ) = s
0
1
Mf (x ) =
2πi
Z
c +i ∞
c −i ∞
ds s
x Df (s )
s
Outline Introduction Crash Course in Number Theory Zeta Function Roots of the Zeta Function Prime Number Theorem Riemann Hy
Dirichlet Series of a Convolution
The Dirichlet series of a Dirichlet convolution h = f ∗ g is
equal to the product of Dirichlet series,
Dh (s ) = Df (s )Dg (s )
The Dirichlet series of a convolution inverse g of a function f is
equal to the reciprocal of the Dirichlet series of f , that is,
Reciprocal of Dirichlet Series
1
= Dg (s )
Df (s )
iff
f ∗g =e
Outline Introduction Crash Course in Number Theory Zeta Function Roots of the Zeta Function Prime Number Theorem Riemann Hy
Euler Product
An arithmetic function f is called multiplicative if f (1) = 1
and for any relatively prime integers a and b
f (ab ) = f (a )f (b ).
The Dirichlet series of a multiplicative function f can be
represented as the product over primes
Euler product


∞
Y 
X
f (p k ) 

1 +

Df (s ) =
ks 
p
p
k =1
Outline Introduction Crash Course in Number Theory Zeta Function Roots of the Zeta Function Prime Number Theorem Riemann Hy
Zeta Function
Riemann zeta function
ζ(s ) = D1 (s ) =
∞
X
1
n=1
ns
,
s = σ + it ,
σ>1
By the convolution inverse one can immediately obtain
∞
X µ(n)
1
= Dµ (s ) =
,
ns
ζ(s )
n=1
σ>1
The Euler product for the zeta function has the form


!−1
∞
Y 
X
1  Y
1



ζ(s ) =
1− s
,
1 +
=
p
p ks 
p
k =1
p
σ>1
Outline Introduction Crash Course in Number Theory Zeta Function Roots of the Zeta Function Prime Number Theorem Riemann Hy
Zeta Function
This means that
log ζ(s ) =
∞ X
X
1 1
k =1 p
k p ks
∞
X
Λ(n) 1
=
,
log n ns
n =1
And, therefore,
∞
X
ζ 0 (s )
Λ(n)
−
= DΛ (s ) =
,
ns
ζ(s )
n =1
σ>1
σ>1
Outline Introduction Crash Course in Number Theory Zeta Function Roots of the Zeta Function Prime Number Theorem Riemann Hy
Integral Representations of Zeta Function
Zeta function is the Mellin transform of the integer part
function [x ]
ζ(s ) = s
∞
Z
σ>1
dx x −s −1 [x ],
1
It can be written in the form
ζ(s ) =
s
s−1
∞
Z
dx x −s −1 {x },
−s
σ>0
1
where {x } = x − [x ] is the fractional part of x, which reveals a
simple pole at s = 1.
Zeta function is directly related to the prime counting functions
log ζ(s ) = s
∞
Z
Z
dx x
1
−s −1
Π(x ) = s
∞
dx
1
π(x )
,
x (x s − 1)
σ>1
Outline Introduction Crash Course in Number Theory Zeta Function Roots of the Zeta Function Prime Number Theorem Riemann Hy
Integral Representations of Zeta Function
More importantly
Mellin Transform of Mertens Function
1
=s
ζ(s )
∞
Z
σ>1
dx x −s −1 M (x ),
1
It is easy to show that
1
ζ(s ) =
Γ(s )
∞
Z
dx x s −1
0
1
ex
−1
,
σ>1
Analytic Continuation of Zeta Function
1
ζ(s ) =
Γ(1 − s )
2πi
Z
C
dz (−z )s
,
z ez − 1
where C is the contour that goes from +∞ + i ε to +∞ − i ε around
the origin in the counterclockwise direction.
Outline Introduction Crash Course in Number Theory Zeta Function Roots of the Zeta Function Prime Number Theorem Riemann Hy
Heat Kernel Representation of Zeta Function
The heat trace on the circle S 1 is given by
θ(t ) = Tr exp(−tD 2 ) =
∞
X
e −n
2
πt
=1+2
n=−∞
∞
X
e −n
n =1
Fundamental Duality (Possion Summation Formula)
1
1
θ(t ) = √ θ
t t
!
It is easy to show
ζ(s ) =
1 πs /2
2Γ s
2
∞
Z
0
dt s /2
t [θ(t ) − 1] ,
t
σ>1
2
πt
Outline Introduction Crash Course in Number Theory Zeta Function Roots of the Zeta Function Prime Number Theorem Riemann Hy
Functional Equation for Zeta Function
By using the duality one can get
)
(
Z
πs /2
1 ∞ dt s /2
1
(1−s )/2
[θ(t ) − 1]
t
+t
ζ(s ) = +
Γ 2s s (s − 1) 2 1 t
which provides an analytic continuation for any s and reveals
a pole at s = 1.
This immediately leads to
Functional Equation
ξ(s ) = ξ(1 − s )
where
1
s
ξ(s ) = s (s − 1)π−s /2 Γ
ζ(s )
2
2
Outline Introduction Crash Course in Number Theory Zeta Function Roots of the Zeta Function Prime Number Theorem Riemann Hy
Roots of the Zeta Function
Zeta function is a meromorphic function with a simple pole at
s = 1.
It has some trivial zeros at the negative even integer points,
that is,
ζ(−2n) = 0 ,
n = 1, 2, . . . .
All non-trivial roots coincide with the roots of the entire
function ξ .
Outline Introduction Crash Course in Number Theory Zeta Function Roots of the Zeta Function Prime Number Theorem Riemann Hy
Non-Trivial Roots of the Zeta Function
There are no roots for σ > 1.
A complex number 1 − ρ is a root if and only if ρ is a root.
The roots are located symmetrically with respect to the
critical line
1
σ= .
2
There are no non-trivial roots for σ < 0.
All roots are located in the critical strip
0 ≤ σ ≤ 1.
If ρ is a root then ρ̄ is also a root.
The roots are located symmetrically with respect to the real
axis.
Outline Introduction Crash Course in Number Theory Zeta Function Roots of the Zeta Function Prime Number Theorem Riemann Hy
Non-Trivial Roots of the Zeta Function
There are infinitely many roots on the critical line.
The number of zeros (with s = σ + it) in the rectangle
0 ≤ σ ≤ 1,
0<t <T
as T → ∞ is
N (T ) =
T
T
7
1
T log
−
+ + O ( log T )
2π
2π 2π
8
There holds
X 1
< ∞,
|ρ|2
ρ
X 1
→∞
|ρ|
ρ
There holds
ψ(x ) = x −
X xρ
ρ
1
1
− log 2π − log 1 − 2
ρ
2
x
!
Outline Introduction Crash Course in Number Theory Zeta Function Roots of the Zeta Function Prime Number Theorem Riemann Hy
Prime Number Theorem
Equivalent formulations of Prime Number Theorem:
As x → ∞
π(x ) ∼ Li(x ),
where Li(x ) =
>x
0
dt
log t
ψ(x ) ∼ x ,
π(x ) ∼
or
x
log x
is the integral logarithm
or
lim
x →∞
X1
ρ
ρ
x ρ−1 = 0
There are no roots of zeta function on the line σ = 1.
Outline Introduction Crash Course in Number Theory Zeta Function Roots of the Zeta Function Prime Number Theorem Riemann Hy
Riemann Hypothesis
The equivalent formulations of the Riemann hypothesis
Riemann Hypothesis
All non-trivial roots of the zeta function are located on the critical
line.
There are no non-trivial roots for σ , 1/2.
π(x ) = Li(x ) + O (x 1/2 log x )
For any ε > 0
For any ε > 0
π(x ) = Li(x ) + O x 1/2+ε
ψ(x ) = x + O x 1/2+ε
Outline Introduction Crash Course in Number Theory Zeta Function Roots of the Zeta Function Prime Number Theorem Riemann Hy
Riemann Hypothesis
For any ε > 0
X1
ρ
The series
ρ
x ρ = O x 1/2+ε
∞
X
µ(n)
n =1
ns
converges for σ > 1/2.
For any ε > 0 the Mertens function M (x ) grows less rapidly
than x 1/2+ε , that is, for any ε > 0
M (x ) = O x 1/2+ε
For large n that is equal to the product of distinct primes the
probablity of even and odd number of distinct primes are
equal.
Outline Introduction Crash Course in Number Theory Zeta Function Roots of the Zeta Function Prime Number Theorem Riemann Hy
Hilbert-Polya Conjecture
Hilbert-Polya Conjecture
There is a unbounded self-adjoint operator H such that the
spectrum of the operator
1
+ iH
2
coincides with the zeros of the Riemann zeta function
Hints:
the dynamics is chaotic, that is, unstable and bounded,
the dynamics is time-irreversible,
periodic orbits have periods independent of energy,
dynamics is quasi-one-dimensional.
Outline Introduction Crash Course in Number Theory Zeta Function Roots of the Zeta Function Prime Number Theorem Riemann Hy
Randomness in Prime Numbers
Suppose an unbiased coin is flipped a large number of times.
Then the probability of getting exactly H heads in N trials is
given by the binomial distribution
!
N 1
B (N , H ) =
H 2N
de Moivre-Laplace Theorem
As N → ∞ the binomial distribution approaches the√normal
distribution with mean N /2 and standard deviation N /2,
r
B (N , H ) ∼ F (N , H ) =

 2
2
N
exp −
H−
πN
N
2
!2 


Outline Introduction Crash Course in Number Theory Zeta Function Roots of the Zeta Function Prime Number Theorem Riemann Hy
Randomness in Prime Numbers
The probability that the number of heads deviates from the
expected value N2 by KN 1/2+ε as N → ∞ is equal
!
N P H − < KN 1/2+ε ∼
2
√
2
= √
π
Z
2 Nε
N /2+
ZKN 1/2+ε
dH F (N , H )
N /2−KN 1/2+ε
2
dx e −x → 1
0
The probability that the number of heads deviates from the
number of tails by 2KN 1/2+ε as N → ∞
P |H − T | < 2KN 1/2+ε → 1
Outline Introduction Crash Course in Number Theory Zeta Function Roots of the Zeta Function Prime Number Theorem Riemann Hy
Randomness in Prime Numbers
With probability 1 the number of heads minus the number of
tails grows less rapidly than N 1/2+ε .
Let n be a large integer such that it is product of distinct
primes, so that, µ(n) , 0.
If the number of factors is odd then µ(n) = −1 and if the
number of factors is even then µ(n) = 1.
Then the evaluation of the Mertens function M (x ) is like
flipping a coin for each integer n ≤ x which is a product of
distinct primes and subtracting the number of tails from the
number of heads.
If the values of µ(n) are independent, that is, with equal
probability a large square-free integer has either an odd
number of factors or an even number of factors, then
for any
given ε > 0 as x → ∞ with probability 1, M (x ) < O x 1/2+ε .
This is equivalent to Riemann hypothesis.
Outline Introduction Crash Course in Number Theory Zeta Function Roots of the Zeta Function Prime Number Theorem Riemann Hy
Random Hermitian Matrices
There are conjectures that relate the statistical behavior of the
nontrivial zeros of the Riemann zeta function to the
distribution of the eigenvalues of large N × N random
Hermitian matrices M with the probability measure
(
N
P (M ) = CN exp − tr M 2
2
)
Such a system is called the Gaussian Unitary Ensemble.
The zeros of the zeta function
ρk =
1
+ itk
2
are distributed like the eigenvalues of the eigenvalues of
random Hermitian matrices.
Outline Introduction Crash Course in Number Theory Zeta Function Roots of the Zeta Function Prime Number Theorem Riemann Hy
Free Riemann Gas
The free Riemann gas is a quantum theory of identical
non-interacting particles.
The particles can be in infinitely many different states labeled
by primes p
The particles can be either bosonic or fermionic
Bosonic particles can be in the same state
Fermionic particles cannot be in the same state (Pauli
Exclusion Principle)
Every state of a system can be described by a positive integer
n=
Y
p kp
p
In such a state there k1 particles in state p1 , k2 particles in
state p2 , etc.
Outline Introduction Crash Course in Number Theory Zeta Function Roots of the Zeta Function Prime Number Theorem Riemann Hy
Free Riemann Gas
States of bosonic particles are described by any integer
States of fermionic particles are described only by integers
which are product of distinct primes, that is, kp are equal to 0
or 1.
The states of fermionic particles can be even or odd
depending on the number of particles, that is, the parity of
the fermionic state is equal to the Möbius function µ(n)
Outline Introduction Crash Course in Number Theory Zeta Function Roots of the Zeta Function Prime Number Theorem Riemann Hy
Free Riemann Gas
The energy of a particle in the state p is log p.
The energy of the state |ni is
E (n) =
X
kp log p = log
p
Y
p kp = log n
p
The partition function of the bosonic system is
Z (β) = Tr exp(−βH ) =
∞
X
exp [−βE (n)] = ζ(β) ,
n =1
where β = 1/T is the inverse temperature.
The partition function of the fermionic system is
Z (β) = Str exp(−βH ) =
∞
X
n =1
µ(n) exp [−βE (n)] =
1
ζ(β)