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1
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
3
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
4
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
5
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
6
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
8
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
12
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
13
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
14
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!
15
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
16
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
17
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
18
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)
19
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
20
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
21
>
+=0Im
2 log)1(2log)()()(
xttt
dtxLixLixJ
here
=
x
t
dtxLi
0log
)(
From:http://mathworld.wolfram.com/LogarithmicIntegral.html
22
>
+=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
+==
===+
23
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
24
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
25
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.
2
1)Re( =s
1 )Re(s
)Im(s
26
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
27
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
28
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
29
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?
30
From: J. Brian Conrey, Notices of the AMS, March 2003, p. 341
Connection between random matrices and (s)
confirmed numerically:
31
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.
32
Background from: http://www.math.ucsb.edu/~stopple/zeta.html
Conclusion
• Complex analysis is impressively powerful
•Profoundly surprising connections:
primes, energy levels, quantum chaos
33
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
34
What to see the slides again?
Can be found at: www.ma.imperial.ac.uk/~hjjens