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1. Introduction. Our goal in this paper is to study the distribution of zeros of the Riemann zeta function as well as of more general L-functions. According to conjectures of Langlands [14-1, the most general L-function is that attached to" an automorphic representation of GLN over a number field, and these in turn should be expressible as products of the "standard" L-functions L(s, r) attached to cuspidal automorphic representations of GLm over the rationals. Such L-functions are therefore believed to be the building blocks for general L-functions, and we call them (principal) primitive L-functions of degree m. (They do not factor as products of such L-functions.) 2 For m 1 these are the Riemann zeta function ((s) and Dirichlet L-functions L(s, 7.) with ;t primitive. For m 2 the analytic properties and functional equation of such L-functions were investigated by Hecke and Maass, and for m > 3 by Godement and Jacquet [5]. We are interested in the fine structure of the distribution of the nontrivial zeros of such primitive L(s, r).Let p= (1/2)+ i denote these zeros. To motivate the formulation of our results, we begin by assuming the Riemann hypothesis (RH) for L(s, r), that is, that ,,0 R. We order the ,t)'s (with multiplicities)The number of y's in an interval I-T, T + 1] is asymptotic to (m/2r)log T as T (see (2.11)). It follows that the numbers = (m/2rr) logical have unit mean spacing. The problem is to understand the statistical nature of the sequence )" Do they come down randomly (Poisson process) or do they follow a more revealing distribution?In the case of the Riemann zeta function, following the original calculation by Montgomery [20] of the pair correlation (see below) and the extensive numerical calculations of Odlyzko [21], [22], it is now well accepted (but far from proven) that the consecutive spacings follow the Gaussian unitary ensemble (GUE) distribution from random matrix theory. That is, if fin n+l Yn are the normalized The reader interested only in the Riemann zeta function '(s) should read the paper with L(s, n) replaced by ((s) and m everywhere, in which case the results were announced in [26]. 2It is quite plausible that these coincide with the primitive Dirichlet series introduced by Selberg[29] or the "arithmetic Dirichlet series" in Piatetski-Shapiro [24].

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Abstract. 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. The Montgomery-Odlyzko LawWe begin with the Riemann Zeta function and some phenomenology associated with it.the product being over the primes, and it converges for Re(s) > 1. As was shown by Riemann [RI] ζ(s) has a continuation to the complex plane and satisfies a functional equationξ(s) is entire except for simple poles at s = 0 and 1. We write the zeroes of ξ(s) asFrom (1) it is clear that |Im(γ)| ≤ 1/2. Hadamard and de la Vallee Poisson in their (independent) proofs of the Prime Number Theorem established that |Im(γ)| < 1/2.