Selberg’s Central Limit Theorem for families of L-functions

Abstract

In this thesis, we present a simple proof of Selberg’s Central Limit Theorem for appropriate families of L-functions. As conjectured by Selberg, his central limit theorem can only be proven for the L-functions belonging to the Selberg Class.

First, we prove Selberg’s central limit theorem for classical automorphic L-functions of degree 2 associated with holomorphic cusp forms. We prove this result in the t-aspect.

In Chapter 4, we prove Selberg’s central limit theorem for Dirichlet L-functions and quadratic Dirichlet L functions associated with primitive Dirichlet characters and twisted Hecke-Maass cusp forms respectively. We prove these results in the q-aspect, i.e., instead of integrating we average over Dirichlet characters.

Finally, in Chapter 5, we prove that a sequence of degree 2 automorphic L-functions attached to a sequence of primitive holomorphic cusp forms form a Gaussian process. Also, any two elements from this sequence of L-functions are pair-wise independent. Additionally, we construct a random matrix that generalizes the notion of independence of the families of automorphic L-functions.