L-functions and the characteristic polynomials of random matrices

Abstract

My purpose in these lecture notes is to review and explain the basic ideas underlying the connection between random matrix theory and the moments of L-functions. Both of these subjects are introduced separately, and at length, in other lectures. I will focus on their intersection – specifically on the way in which random matrix theory can be used to predict values of the moments and on some applications of the resulting conjectures to other important problems in number theory. The ideas I shall be reviewing were introduced in [17], [18] and [3], and the applications in [8] and [9]. More recent results and developments, such as those related to calculating complete asymptotic expansions of the moments [4, 5], are described in other lecture notes.