Nicholas M. Katz, Peter Sarnak
The spacing distributions between zeros of zeta functions
(222K, amstex)
ABSTRACT. In a remarkable numerical experiment, Odlyzko has found
that the local spacing distribution between the zeros of the Riemann
Zeta function is modelled by the eigenvalue distributions coming from
random matrix theory. In particular by the ``GUE''
(Gaussian Unitary Ensemble) model. His experiment was inspired by the
paper of Montgomery who determined the pair correlation distribution
for the zeros (in a restricted range). We will refer to the above
phenomenon as the Montgomery--Odlyzko Law. Rudnick and
Sarnak have determined the $n \geq 2$ correlations
for the zeros of the zeta function, as well as for more
general automorphic $L$--functions (again only in
restricted ranges). These are in perfect agreement with GUE predictions.
It appears that the Montgomery--Odlyzko Law is a universal feature for
such $L$--functions. However, a complete proof of this law is well beyond
the range of existing techniques. If one believes that the above
phenomenon is a manifestation of the spectral nature of the zeros, then it
is natural to ask if there is such a law for the zeta and $L$--functions
associated to curves and exponential sums over finite fields. For, in
these cases, their zeros may be realized as eigenvalues of Frobenius on
cohomology groups. One of the goals of this paper is the
formulation and proof of an analogue of the Montgomery--Odlyzko Law for
these zeta and $L$--functions.