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. 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, an