Selberg’s Central Limit Theorem for families of L-functions
Das, MD
Date: 16 January 2023
Thesis or dissertation
Publisher
University of Exeter
Degree Title
MSc by Research in Mathematics
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 ...
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.
MbyRes Dissertations
Doctoral College
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