Random matrix theory and L-functions: Zeros of symmetric power L functions

We give a general view of the links between random matrix theory and L-functions. Results on the average one-level density of the zeros of L-functions of modular forms, due to Iwaniec, Luo & Sarnak, are given. We explain the Mock-Gaussian behavior (first discovered by Hughes & Rudnick) in the case of L-functions of modular forms, a computation of the moments due to Miller & Rudncik. Finally we give results obtained with Ricotta on the one and two level densities for symmetric power L-functions.

Number Theory Seminar

Warwick Mathematical Institute, The University of Warwick, Warwick, Royaume-Uni.