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,
Complex functions,
Energy levels and
Riemann
Henrik Jeldtoft Jensen,
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
= ei
Energy levels of
heavy nucleuses
Dyson
Zeroes of (s)
Quantum mechanics of
chaotic systems
Riemann
Hugh L Montgomery
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Odlyzko
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
8
6
4
2
0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17
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Gauss’ estimate:
x
( x) log( x )
From J Derbyshire: Prime Obsession
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Riemann’s analytic approach.
The zeta function:
( s) = n
s
n =1
Examples:
2
1 1 1
( 2) = 1 + + + + ... =
4 9 16
6
1 1 1 1
(1) = 1 + + + + + ... = 2 3 4 5
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Domain of definition:
iy
s=x+iy
x
1
1
1
1
( s ) = s s = x +iy
n =1 n
n =1 n
n =1 n
=
n =1
1
1
= x
x
iy
n n
n =1 n
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n
x
n
1
2
Area under blue steps
<
Area under red curve
1
x
<
1
+
n
dn < x
2
n =1 n
When x>1
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And what about the primes?
1 ( s ) = 1 s p p prime According to Euler:
1
1
because:
then:
1 1
1
1
1 s = 1 + s + 2 s + 3s + ...
p p
p
p
1
(p
m1
1
m2
2
p Lp
)
mq s
q
1
= s
n
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From Euler’s product we learn:
1. Prime numbers are hidden inside
2. And
(s )
( s ) 0 when product formula valid
Since
1
1
1
1 0 = 1 s p such that 1 - s = 0
p p p Not possible
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Summary so far
(s )
Structure of
(s ) :
( s ) 0 for Re( s ) > 1
s=1
Where is the relation to the primes?
Can one make sense of
(s ) for Re(s)<1?
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We start with the Re(s) <1 issue.
To illustrate the nature of what Riemann did consider the
function:
1
1 - z for z < 1
2
3
f ( z) = 1 + z + z + z + L = for z < 1
Cannot be used for
|z|>1
Well defined for all z 1
The extension beyond |z|<1 is UNIQUE!
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Riemann’s extension of (z )
Im(s )
(1 s ) u s 1
du
( s) =
u
2i P e 1
Re(s )
P
Valid for all z1
Where (z ) is the Gamma
function originating from the
usual n! = n ( n 1)( n 2) L 2 1
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The functional equation for the zeta function
(
2 ) sin( s / 2) (1 s )
( s) =
s
sin( s ) ( s )
From this follows that
( s ) = 0 for s = 2,4,6,K
and that all other zeroes MUST lie in the strip
0 Re( s ) 1
the critical strip
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Why is it interesting?
Because the primes can be found from the zeroes of (s).
Consider
0 for x = 0
jump by 1 when x = prime p
J ( x ) = jump by 1 when x = prime squares p 2
2
1 when x = prime cubes p 3
jump
by
3
ect.
J(x)
Defined to be “halfway” at
jumps
1
2
x
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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)
Procedure:
complex zeroes
of (x)
J(x)
(x)
&
Li(x)
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Riemann’s hypothesis
All complex zeroes are confined to
1
Re( s ) =
2
With
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( s) = n s
20
n =1
continued to
10
-2
1
(1 s ) u s 1
du
( s) =
u
2i P e 1
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Why is it interesting?
Because the primes can be found from the zeroes of (s).
According to Riemann:
M
μ (n)
J ( x1 / n ),
( x) = n
n =1
with μ = 1,0,1 and M = log x
log 2
and
dt
J ( x ) = Li ( x ) Li ( x ) log 2 + 2
t (t 1) log t
Im >0
x
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here
dt
J ( x ) = Li ( x ) Li ( x ) log 2 + 2
t (t 1) log t
Im >0
x
x
dt
log t
0
Li ( x ) = From:http://mathworld.wolfram.com/LogarithmicIntegral.html
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dt
J ( x ) = Li ( x ) Li ( x ) log 2 + 2
t (t 1) log t
Im >0
x
The sum over the zeroes contains oscillatory terms
since
x =x
1
+it
2
= xe
= x x = x (e
it
it ln x
)
ln x it
= x [cos(t ln x ) + i sin(t ln x )]
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Adding up the contribution from the zeroes
From (http://www.maths.ex.ac.uk/~mwatkins/zeta/encoding2.htm)
(x)
Some zeroes included
x
No zeroes included
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Hunting the zeroes
From functional equation
s
(
2 ) sin( s / 2) (1 s )
( s) =
sin( s ) ( s )
Follows that all zeroes must be confined in the
strip
0 Re( s ) 1
Im(s )
1
Re(s )
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Riemann conjectured in 1859 that zeroes are all on
the line
1
Re( s ) =
2
But he couldn’t prove it!
Im(s )
1
Re(s )
Nor has anyone else been able to ever since. Now
it is the Riemann Hypothesis.
Prove it and get 106 $ from the Clay Institute.
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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:
J ( x ) = Li ( x ) Im >0
dt
2
t
(
t
1) log t
x
Li ( x ) log 2 + x Re( s ) [cos(t ln x ) + i sin(t ln x )]
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
sin u 1 u 2
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Dyson spotted that the same function describes the
spacing between energy levels, i.e., eigenvalues of
Hamiltonians describing big nucleuses:
Im(s)
E
Statistically equivalent?
Re(s)
Random matrices
(Gaussian unitary ensemble)
1/2
The imaginary zeroes of (s)
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Connection between random matrices and (s)
confirmed numerically:
From: J. Brian Conrey, Notices of the AMS, March 2003, p. 341
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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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Conclusion
• Complex analysis is impressively powerful
•Profoundly surprising connections:
primes, energy levels, quantum chaos
Background from:
http://www.math.ucsb.edu/~stopple/zeta.html
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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
A very useful link that contains a wealth of references:
Matthew R. Watkins' home page namely:
http://www.maths.ex.ac.uk/~mwatkins/
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What to see the slides again?
Can be found at: www.ma.imperial.ac.uk/~hjjens
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