Fonction $\zeta$ et matrices aléatoires

Abstract

Since Euler and Riemann, various links have been established between the behaviour of prime numbers and the analytical properties of the Riemann ζ function. The zeroes of ζ have a great importance that justifies their fine study. Since seventies, a rich tool has appeared for the study of zeroes: the statistical spectral properties of the unitary matrices that are a model for ζ. The aim of this survey is to explain the link between unitary matrices and ζ, and its extension to the L-functions.