Hilbert and Polya suggested that there might be a natural spectral interpretation of the zeroes of the Riemann Zeta function. While at the time there was little evidence for this, today the evidence is quite convincing. Firstly, there are the “function field” analogues, that is zeta functions of curves over finite fields and their generalizations. For these a spectral interpretation for their zeroes exists in terms of eigenvalues of Frobenius on cohomology. Secondly, the developments, both theoretical and numerical, on the local spacing distributions between the high zeroes of the zeta function and its generalizations give striking evidence for such a spectral connection. Moreover, the low-lying zeroes of various families of zeta functions follow laws for the eigenvalue distributions of members of the classical groups. In this paper we review these developments. In order to present the material fluently, we do not proceed in chronological order of discovery. Also, in concentrating entirely on the subject matter of the title, we are ignoring the standard body of important work that has been done on the zeta function and L-functions.

Comparison between formulae for the counting functions of the heights t n of the Riemann zeros and of semiclassical quantum eigenvalues En suggests that the t n are eigenvalues of an (unknown) hermitean operator H, obtained by quantizing a classical dynamical system with hamiltonian H cl . Many features of H cl are provided by the analogy; for example, the "Riemann dynamics" should be chaotic and have periodic orbits whose periods are multiples of logarithms of prime numbers. Statistics of the t n have a similar structure to those of the semiclassical En ; in particular, they display random-matrix universality at short range, and nonuniversal behaviour over longer ranges. Very refined features of the statistics of the t n can be computed accurately from formulae with quantum analogues. The Riemann-Siegel formula for the zeta function is described in detail. Its interpretation as a relation between long and short periodic orbits gives further insights into the quantum spectral fluctuations. We speculate that the Riemann dynamics is related to the trajectories generated by the classical hamiltonian H cl = XP. Key words. spectral asymptotics, number theory AMS subject classifications. 11M26, 11M06, 35P20, 35Q40, 41A60, 81Q10, 81Q50 PII. S0036144598347497 1.

By looking at the average behavior (n-level density) of the low lying zeros of certain families of L-functions, we find evidence, as predicted by function field analogs, in favor of a spectral interpretation of the non-trivial zeros in terms of the classical compact groups. This is further supported by numerical experiments for which an efficient algorithm to compute L-functions was developed and implemented. iii Acknowledgements When Mike Rubinstein woke up one morning he was shocked to discover that he was writing the acknowledgements to his thesis. After two screenplays, a 40000 word manifesto, and many fruitless attempts at making sushi, something resembling a detailed academic work has emerged for which he has people to thank. Peter Sarnak- from Chebyshev's Bias to USp(1). For being a terrific advisor and teacher. For choosing problems suited to my talents and involving me in this great project to understand the zeros of L-functions. Zeev Rudnick and Andrew Oldyzko for many disc...

. Let E=Q(T ) be a one-parameter family of elliptic curves. Assuming various standard conjectures, we give an upper bound for the average rank of the fibers E t (Q) with t 2 Z, improving earlier estimates of Fouvry-Pomykala and Michel. We also show how certain assumptions about the distribution of zeros of L-series might help explain the experimentally observed fact that the average rank of the fibers appears to be strictly larger than the naive expected value of rank E(Q(T )) + 1=2. Introduction Let E ! P 1 be an elliptic surface defined over Q , say given concretely by a Weierstrass equation E : y 2 + a 1 (T )xy + a 3 (T )y = x 3 + a 2 (T )x 2 + a 4 (T )x + a 6 (T ) with a i (T ) 2 Z[T ] and discriminant \Delta(T ) 6= 0. We also let N(T ) denote the conductor polynomial of E , that is, N(T ) = Y \Delta(ff)=0 (T \Gamma ff) \Theta Y c4 (ff)=c6 (ff)=0 (T \Gamma ff); where c 4 (T ) and c 6 (T ) are the usual quantities associated to the Weierstrass equation for E . Ther...

29> E(K). Clearly E(K) = [ t2C(K) E t (K): A key to studying rational points on elliptic surfaces is to exploit the fact that the fibers are elliptic curves, hence have a group structure. Note that if g(C) 2, then E(K) lies on a finite union of fibers, so the problem of describing E(K) reduces to the two problems of describing the finitely many points in C(K), and the group of rational points in E t (K) for each t 2 C(K). Thus the problem is, in some sense, one-dimensional (although by no means easy, as will be explained in Poonen's talk). We will generally assume that<F65.2