97-480 Nicholas M. Katz, Peter Sarnak

The spacing distributions between zeros of zeta functions (222K, amstex) Sep 9, 97

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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.

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