The Spacing Distributions between Zeros of Zeta Functions

0. Introduction In a remarkable numerical experiment, Odlyzko Od] has found that the local spacing distribution between the zeros of the Riemann Zeta function is modelled by the eigen-value distributions coming from random matrix theory. In particular by the \GUE" (Gaussian Unitary Ensemble) model Gau]. His experiment was inspired by the paper of Montgomery Mon] 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 Ru{Sa] have determined the n 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 nite elds. For, in these cases, their zeros may be realized as eigenvalues of Frobenius on cohomology groups (see Section 7). 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. A major part of this work is devoted to an analysis of the distribution of the local spacings between the eigenvalues of matrices in the classical groups. Let G(N) denote one of the compact groups U