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<h1 align=center> Physics and zeros of the zeta-function
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The local statistics of the imaginary parts $\gamma_n$ of the
complex zeros of the Riemann zeta function show the characteristic
distribution seen in the eigenvalues of a matrix pulled at random
from the unitary group endowed with Haar measure.  This collection
of matrices is called by physicists the CUE: circular unitary
ensemble.  For more details on the statistics of the zeros of the
zeta function and other $L$-functions, see [\begin{rawhtml}<a href="http://aimath.org/WWN/lrmt/">L-functions and
Random Matrix Theory</a>\end{rawhtml}].

These very same CUE statistics are also seen on  a local scale
when one studies the distribution of the semiclassical eigenvalues
$E_n$ of quantum systems having classical analogues that display
chaotic behaviour and are not symmetric under time-reversal.

This suggests that the $\gamma_n$ can be construed as the
eigenvalues of some Hermitean operator which is itself obtained by
quantizing a classical dynamical system sharing the properties
mentioned above: chaoticity and no time-reversal symmetry.

Furthermore, the long-range statistics of $\gamma_n$ for the
Riemann zeta function depend on the prime numbers in a manner
which is very accurately predicted by formulae with analogues in
semiclassical periodic orbit theory.  This suggests that the
periodic orbits of the hypothetical system underlying the Riemann
zeta function would be determined by the positions of the prime
numbers.

Clearly the identification of a dynamical system which when
quantized produced a Hermitean operator with eigenvalues $E_n$
related to the complex Riemann zeros $\rho_n$ by $iE_n=\rho_n-1/2$
would lead to a proof of the Riemann Hypothesis.  In the field of
quantum chaos studies are made of the very systems which are
relevant in such a search, and results from this field suggest
what many of the characteristics of such a system should be.  For
a detailed review of these issues and further references see Berry
and Keating [
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<a href="http://www.ams.org/msnmain?fn=130&s1=2000f:11107&pg1=MR">MR 2000f:11107</a>
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Back to the 
<a href="http://www.aimath.org/WWN/rh/"> main index</a> 
for The Riemann Hypothesis.
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