Random matrices and L-functions

In recent years there has been a growing interest in connections between the statistical properties of number theoretical L-functions and random matrix theory. We review the history of these connections, some of the major achievements and a number of applications. PACS numbers: 02.10.De, 02.10.Yn 1. The history in brief Number theory and random matrix theory met, by chance, over a cup of tea in the common room at the Institute for Advanced Study in Princeton in the early 1970s. H L Montgomery, then a graduate student, had shown his latest work to F J Dyson. His main conclusion was a conjecture for the two-point correlation function of the zeros of the Riemann zeta function, which he had managed to prove for a limited range of correlations [57]. Dyson recognized this to be the same as the two-point correlation function, which he had calculated a decade earlier, for the eigenvalues of matrices drawn at random from U(N) (the group of N × N unitary matrices) uniformly with respect to Haar measure [30]. The Riemann zeta function is extremely important in number theory because it allows for analytical techniques to be applied to the study of the distribution of prime numbers, and Montgomery’s conjecture plays a central role in the theory of this distribution. Although the full proof of Montgomery’s conjecture has still escaped completion, Odlyzko’s numerical computations have provided very strong evidence in support of it [59]. Working on the powerful computers of AT&T, in the 1980s Odlyzko calculated batches of the Riemann zeros high up on the critical line where the Riemann hypothesis places them, and computed numerically the two-point correlation function, as well as many other statistics of the zeros. He also computed the distribution of the values of the zeta function. His numerics provide very convincing evidence that, as predicted by Montgomery’s conjecture, the two-point correlation function of the Riemann zeros converges to the random matrix result of Dyson as zeros higher and higher on the critical line are considered. Odlyzko’s numerical work continues today and he is currently working at the dizzying height of the 1022nd Riemann zero. 0305-4470/03/122859+23$30.00 © 2003 IOP Publishing Ltd Printed in the UK 2859 2860 J P Keating and N C Snaith In the 1980s the Riemann zeta function also took on a new life in mathematical and theoretical physics when it became a tool in the field of quantum chaos. The eigenvalues of complex quantum systems (e.g., nuclei [4, 18, 35] and disordered mesoscopic systems [31]) and systems showing chaotic behaviour in the classical limit [12, 16, 17] display the statistics of the three standard ensembles of Hermitian matrices, the Gaussian unitary ensemble (GUE), the Gaussian orthogonal ensemble (GOE) and the Gaussian symplectic ensemble (GSE), or equivalently (in the appropriate limit) the three corresponding standard ensembles of unitary matrices: the CUE, COE and CSE. (Here the ‘C’ stands for ‘circular’.) However, while the eigenvalue statistics show certain behaviour which is easily identified and predicted by random matrices, there are further characteristics which are system-specific and are connected to the periodic orbits of the system under consideration [6]. These relate to the approach to the random matrix limit as h̄ → 0. It was realized by Berry [7] that very similar contributions result from the primes and behave in the same way with respect to the statistics of the Riemann zeros as do the periodic orbits to the statistics of the eigenvalues. In this case, the primes describe the approach to the random-matrix limit as the height up the critical line increases. This launched an era of study of the Riemann zeta function in the field of quantum chaos [2, 6, 8–11, 47]. Most of the results are reviewed in [11, 48]. In essence, the connection between the Riemann zeta function and the prime numbers was being used to point the way through more complicated periodic orbit calculations; and in return the familiarity which arose with the zeta function enabled physicists to contribute insights from physics to its study. In particular, it resulted in a return to the question of the statistics of the Riemann zeros when Bogomolny and Keating [13, 15] in 1995 and 1996 showed, subject to certain conjectures of Hardy and Littlewood concerning the distribution of primes, that not just the two-point, but the general n-point statistics of the Riemann zeros are the same as those of the eigenvalues of random unitary matrices in the limit as one looks at zeros infinitely high up the critical line. At the same time, first Hejhal [37] with the three-point case, then Rudnick and Sarnak [61] generalized Montgomery’s theorem by proving for a limited range of correlations, as in the two-point statistic, that the n-point correlations of the Riemann zeros high on the critical line coincide with the corresponding random unitary matrix statistics. At the end of the 1990s, two developments occurred which illustrate how deeply random matrix theory is intertwined with the Riemann zeta function. Far from there being some accidental similarity between the zeros of this one function and the eigenvalues of random matrices, it became apparent that this connection was far more general. The zeta function is but one example of a broader class of functions known as L-functions. These all satisfy generalizations of the Riemann hypothesis. For any individual L-function, it is believed that the zeros high up on the critical line are distributed like the eigenvalues of random unitary matrices, that is, exactly as in the case of the Riemann zeta function [57, 61]. More interesting, however, is the fact that it has been conjectured by Katz and Sarnak [45, 46] that averages over various families of L-functions, with the height up the critical line of each one fixed, are described not only by averages over the unitary group U(N), but by averages over other classical compact groups, for example the orthogonal group O(N) or the unitary symplectic group USp(2N), depending upon the family in question. The eigenvalue statistics of these groups have also been found to occur in disordered superconducting systems [1]. The second of the two significant recent developments was the discovery that calculations purely within random matrix theory can suggest answers to important questions that number theorists have been unable to make progress on using standard techniques. In [51, 52], Keating and Snaith showed that by studying the value distribution and moments of the characteristic polynomial of a random matrix, one can make predictions about the value distribution and moments of the Riemann zeta function and other L-functions. The reason for this is clear: the Random matrices and L-functions 2861 characteristic polynomial of a random matrix has zeros (the eigenvalues of the matrix) which, conjecturally, show the same statistical behaviour of the zeros of L-functions. Thus random matrix theory can be put to very practical use in the study of L-functions. Our aim in the following sections is to expand further on the developments outlined above. Specifically, we will concentrate on those aspects not covered in previous reviews [11, 48]. 2. Pair correlations We begin with some basic facts about the Riemann zeta function. The Riemann zeta function is defined by