Number Theory: An Introduction to Mathematics

384 IX The Number of Prime Numbers

In the language of physics Montgomery’s conjecture says that 1−(sinπu/πu)^2
is thepair correlation functionof the zeros ofζ(s). Dyson pointed out that this is
also the pair correlation function of the normalized eigenvalues of a randomN×N
Hermitian matrix in the limitN→∞. A great deal more is known about this so-
calledGaussian unitary ensemble, which Wigner (1955) used to model the statistical
properties of the spectra of complex nuclei. For example, if the eigenvalues are nor-
malized so that the average difference between consecutive eigenvalues is 1, then the
probability that the difference between an eigenvalue and the least eigenvalue greater
than it does not exceedβconverges asN→∞to

∫β

0

p(u)du,

where the density functionp(u)can be explicitly specified.
It has been further conjectured that the spacings of the normalized zeros of the
zeta-function have the same distribution. To make this precise, let the zeros 1/ 2 +iγn
ofζ(s)withγn>0 be numbered so that

γ 1 ≤γ 2 ≤···.

Since it is known that the number ofγ’s in an interval [T,T+1] is asymptotic to
(logT)/ 2 πasT→∞, we put

γ ̃n=(γnlogγn)/ 2 π,

so that the average difference between consecutiveγ ̃nis 1. Ifδn= ̃γn+ 1 − ̃γn,andif
vN(β)is the number ofδn≤βwithn≤N, then the conjecture is that for eachβ> 0

vN(β)/N→

∫β

0

p(u)du asN→∞.

This nearest neighbour conjecture and the Montgomery pair correlation conjecture
have been extensively tested by Odlyzko (1987/9) with the aid of a supercomputer.
There is good agreement between the conjectures and the numerical results.

5 Generalizations and Analogues

The prime number theorem may be generalized to any algebraic number field in the
following way. LetKbe an algebraic number field, i.e. a finite extension of the field
Qof rational numbers. LetRbe the ring of all algebraic integers inK,Ithe set of all
nonzero ideals ofR,andPthe subset of prime ideals. For anyA∈I, the quotient
ringR/Ais finite; its cardinality will be denoted by|A|and called thenormofA.
It may be shown that theDedekind zeta-function

ζK(s)=

∑

A∈I

|A|−s

is defined forRs>1 and that the product formula