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Subtle relations: prime numbers, complex functions, energy

levels and Riemann.

Prof. Henrik J. Jensen, Department of Mathematics, Imperial College London.

At least since the old Greeks mathematics the nature of the prime numbers has

puzzled people. To understand how the primes are distributed Gauss studied the

number (x) of primes less than a given number x. Gauss fund empirically that (x)

is approximately given by x/log(x). In 1859 Riemann published a short paper where

he established an exact expression for (x). However, this expression involves a

sum over the zeroes of a certain complex function. The properties of the zeroes out

in the complex plane determine the properties of the primes! Riemann conjectured

that all the relevant zeroes have real part . This has become known as the

Riemann hypothesis and is considered to be one of the most important unsolved

problems in mathematics. Some statistical properties of the zeroes are known.

Surprisingly it has become clear that the zeroes are distributed according to the

same probability function that describes the energy levels of big atomic nucleuses.

Hence we arrive at a connection between the prime numbers and quantum energy

levels in nuclear physics.

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Subtle relations: Prime numbers,Subtle relations: Prime numbers,

Complex functions, Complex functions,

Energy levels and Energy levels and

Riemann Riemann

Henrik Jeldtoft Jensen,Henrik Jeldtoft Jensen,

Dept of Mathematics Dept of Mathematics

Background from: http://www.math.ucsb.edu/~stopple/zeta.html

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Even numbers: 2, 4, 6, 8 , …

Odd numbers: 1, 3, 5, 7, …

The primes: 2, 3, 5, 7, 11, 13, … what’s the pattern?

Complex numbers

s=x+iy

= ie

Energy levels of

heavy nucleuses

Quantum mechanics of

chaotic systems Zeroes of (s)

Riemann

Dyson

Hugh L MontgomeryOdlyzko

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Motto of the talk:

Why it is a good idea to know about

complex numbers and complex

functions.

Copied from http://www.union.ic.ac.uk/dsc/physoc/index.php

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Content:

1. Number of primes less than x

2. Riemann’s 1859 paper

3. Analytic continuation

4. Energy levels of large nucleuses

5. Random matrix theory

6. The same statistics

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Number theory:

(x) = number of primes less than x

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

0

2

4

6

8

7

Gauss’ estimate:)log(

)(x

xx

From J Derbyshire: Prime Obsession

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Riemann’s analytic approach.

The zeta function:

Examples:

=

=1

)(n

sns

=+++++=

=++++=

...5

1

4

1

3

1

2

11)1(

6...

16

1

9

1

4

11)2(

2

9

Domain of definition:

1

s=x+iy

x

iy

==

=+

==

==

==

11

111

11

111)(

n

x

niyx

niyx

n

s

n

s

nnn

nnns

10

1 2

Area under blue steps

n

< Area under red curve

=

<+<

2 1

11

dnnn

x

n

x

When x>1

xn

11

And what about the primes?

According to Euler:

because:

then:

1

11)( =

s

primep ps

...111

11

132

1

++++=ssss

pppp

( ) ssm

q

mm nppp q

11

21

21

=L

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From Euler’s product we learn:

1. Prime numbers are hidden inside

2. And when product formula valid

Since

)(s

0)(s

0p

1-1 that such

110

1

s

1

== ppp

s

Not possible

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Summary so far

Structure of :)(s

s=1

)(s

Where is the relation to the primes?

Can one make sense of for Re(s)<1?)(s

1)Re(for 0)( >ss

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1 for

1for z-1

1

1)( 32

<

<

=++++=

z

z

zzzzf L

We start with the Re(s) <1 issue.

To illustrate the nature of what Riemann did consider the

function:

Cannot be used for

|z|>1 Well defined for all 1z

The extension beyond |z|<1 is UNIQUE!

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Riemann’s extension of )(z

=P

u

s

due

u

i

ss

12

)1()(

1

Where is the Gamma

function originating from the

usual

)(z

12)2)(1(!= Lnnnn

Valid for all z 1

P

)Re(s

)Im(s

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The functional equation for the zeta function

( ))() sin(

)1()2/ sin(2)(

ss

sss

s

=

From this follows that

and that all other zeroes MUST lie in the strip

K,6,4,2for 0)( == ss

1)Re(0 s the critical strip

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Why is it interesting?

Because the primes can be found from the zeroes of (s).

Consider

=

=

=

=

=

.

cubes prime when3

1by jump

squares prime when2

1by jump

prime when1by jump

0for 0

)(

3

2

ect

px

px

px

x

xJ

2

1

x

J(x)

Defined to be “halfway” at

jumps

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Express (x) in terms of J(x)

express J(x) in terms of Li(x)

evaluated at the complex zeroes of

(x)

complex zeroes

of (x)

&

Li(x)

Procedure:

J(x) (x)

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Riemann’s hypothesis

All complex zeroes are confined to

2

1)Re( =s

1-2

10

20

30

=P

u

s

due

u

i

ss

12

)1()(

1

=

=1

)(n

sns

With

continued to

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Why is it interesting?

Because the primes can be found from the zeroes of (s).

According to Riemann:

,)()(

)(1

/1

=

=M

n

nxJ

n

nx

μ

and

>

+=0Im

2 log)1(2log)()()(

xttt

dtxLixLixJ

==2log

log and 1,0,1 with

xMμ

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>

+=0Im

2 log)1(2log)()()(

xttt

dtxLixLixJ

here

=

x

t

dtxLi

0log

)(

From:http://mathworld.wolfram.com/LogarithmicIntegral.html

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>

+=0Im

2 log)1(2log)()()(

xttt

dtxLixLixJ

The sum over the zeroes contains oscillatory terms

since

( )

[ ])lnsin()lncos( ln

ln2

1

xtixtxex

exxxxx

xit

itxit

it

+==

===+

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Adding up the contribution from the zeroes

From (http://www.maths.ex.ac.uk/~mwatkins/zeta/encoding2.htm)

x

(x)

No zeroes included

Some zeroes included

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Hunting the zeroes

From functional equation

Follows that all zeroes must be confined in the

strip

( ))() sin(

)1()2/ sin(2)(

ss

sss

s

=

1)Re(0 s

1 )Re(s

)Im(s

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Riemann conjectured in 1859 that zeroes are all on

the line

But he couldn’t prove it!

Nor has anyone else been able to ever since. Now

it is the Riemann Hypothesis.Riemann Hypothesis.

Prove it and get 106 $ from the Clay Institute.

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1)Re( =s

1 )Re(s

)Im(s

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Numerical results support the Riemann Hypothesis:

• Andrew M. Odlyzko

The 1020-th zero of the Riemann zeta function

and 175 million of its neighbours

• Sebastian Wedeniwski

The first 1011 nontrivial zeros of the Riemann zeta

function lie on the line Re(s) = 1/2 . Thus, the Riemann

Hypothesis is true at least for all

|Im(s)| < 29, 538, 618,432.236

which required 1.3 . 1018 floating-point operations

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A zero off the critical line would induce a pattern in

the distribution of the primes:

>

+=0Im

2 log)1(2log)()()(

xttt

dtxLixLixJ

[ ])lnsin()lncos()Re(xtixtx

s +

Zeroes with different Re(s) would contribute

with different weights

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The distribution along the critical line determines the

distribution of the primes.

So how are they distributed?

Hugh L Montgomery’s Pair Correlation Conjecture:

Derived from the Riemann Hypothesis plus

conjectures concerning twin primes.

Spacing between zeroes controlled by

2

sin1

u

u

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Dyson spotted that the same function describes the

spacing between energy levels, i.e., eigenvalues of

Hamiltonians describing big nucleuses:

E Im(s)

Random matrices

(Gaussian unitary ensemble) The imaginary zeroes of (s)

Re(s)

1/2

Statistically equivalent?

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From: J. Brian Conrey, Notices of the AMS, March 2003, p. 341

Connection between random matrices and (s)

confirmed numerically:

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Puzzle:

However as one study the correlations between the

Nth and the Nth+K zeroes for large N and K the

statistics doesn’t exactly fit the GUE.

M. Berry pointed out that the correlations between

the zeroes of (s) are like the correlations of the

energy levels of a Quantum Chaotic system.

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Background from: http://www.math.ucsb.edu/~stopple/zeta.html

Conclusion

• Complex analysis is impressively powerful

•Profoundly surprising connections:

primes, energy levels, quantum chaos

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A very useful link that contains a wealth of references:

Matthew R. Watkins' home page namely:

http://www.maths.ex.ac.uk/~mwatkins/

Some references:

Technical:

H.M. Edwards: Riemann’s Zeta Function

M.L. Mehta: Random Matrices

Non-technical:

J. Derbyshire: Prime Obsession

M. du Sautoy: The Music of the Primes

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What to see the slides again?

Can be found at: www.ma.imperial.ac.uk/~hjjens