For ease of notation we will phrase everything in terms of
the Riemann -function, with the understanding that
all statements hold for general
-functions with the
obvious modifications.
Assume the Riemann Hypothesis, let
denote
a nontrivial zero of the
-function with
,
and write
for the
th zero, ordered with
increasing imaginary part and repeated according to their
multiplicity.
Let
,
so that
has
average value of 1. An important problem is establishing
various statistical properties of the sequence of
.
The pioneering work of Montgomery [49 #2590] on the pair
correlation of zeros of the -function, work of
Hejhal [
MR 96d:11093], and Rudnick and Sarnak [
MR 97f:11074],
on higher correlations of zeros, and extensive numerical
calculations of Odlyzko (see [
MR 88d:11082] and
[unpublished work]
available on his web page), give persuasive evidence of the
following:
The GUE Conjecture The (suitably rescaled)
zeros of the Riemann -function
are distributed like the eigenvalues of large random matrices from
the Gaussian Unitary Ensemble.
The conjecture has yet to be stated in a precise form.
See The GUE hypothesis for a discussion.
However, any reasonable form of the conjecture makes a
prediction of
the correlation functions of the zeros and
the distribution of the neighbor spacings of the zeros,
as well as various other statistics. Thus, the GUE conjecture is a
powerful tool for making and testing conjectures about the
-function, and for shedding new light on a variety of
long-standing questions. See the discussions
of mean values and value distribution for more
examples of how this conjecture relates to classical objects
studied in number theory.
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for L-functions and Random Matrix Theory.