On random matrices and L-functions

Evidence for deep connections between number theory and random matrix theory has been noticed since the Montgomery-Dyson encounter in 1972 : the function fields case was studied by Katz and Sarnak, and the moments of the Riemann zeta function along its critical axis were conjectured by Keating and Snaith, in connection with similar calculations for randommatrices on the unitary group. This thesis concentrates on the latter aspect : it aims first to give further evidence for this analogy in the number field case, second to develop probabilistic tools of interest for number theoretic questions. The introduction is a survey about the origins and limits of analogies between random matrices in the compact groups and L-functions. We then state the main results of this thesis. The first two chapters give a probabilistic flavor of results by Keating and Snaith, previously obtained by analytic methods. In particular, a common framework is set in which the notion of independence naturally appears from the Haar measure on a compact group. For instance, if g is a random matrix from a compact group endowed with its Haar measure, det(Id − g) may be decomposed as a product of independent random variables. Such independence results hold for the Hua-Pickrell measures, which generalize the Haar measure. Chapter 3 focuses on the point process induced on the spectrum by these laws on the unit circle : these processes are determinantal with an explicit kernel, called the hypergeometric kernel. The universality of this kernel is then shown : it appears for any measure with asymmetric singularities. The characteristic polynomial of random matrices can be considered as an orthogonal polynomial associated to a spectral measure. This point of view combined with the widely developed theory of orthogonal polynomials on the unit circle yields results about the (asymptotic) independence of characteristic polynomials, a large deviations principle for the spectral measure and limit theorems for derivatives and traces. This is developed in Chapters 4 and 5. Chapter 6 concentrates on a number theoretic issue : it contains a central limit theorem for log ζ evaluated at distinct close points. This implies correlations when counting the zeros of ζ in distinct intervals at a mesoscopic level, confirming numerical experiments by Coram and Diaconis. A similar result holds for random matrices from the unitary group, giving further insight for the analogy at a local scale. v te l-0 03 73 73 5, v er si on 2 31 D ec 2 01 0