Weighted central limit theorems for central values of $L$-functions

TL;DR

This paper proves weighted central limit theorems for the distribution of central values of Dirichlet and Hecke $L$-functions, revealing their probabilistic behavior as the modulus grows large, under GRH.

Contribution

It establishes new weighted central limit theorems for $L$-values of Dirichlet characters and Hecke eigenforms, extending probabilistic understanding of these values.

Findings

Weighted CLT for Dirichlet $L$-values as $q o obreak \infty$

Joint distribution CLT for twists of two Hecke eigenforms under GRH

Positive proportion of twists with large or small central $L$-values

Abstract

We establish a central limit theorem for the central values of Dirichlet $L$-functions with respect to a weighted measure on the set of primitive characters modulo $q$ as $q \rightarrow \infty$. Under the Generalized Riemann Hypothesis (GRH), we also prove a weighted central limit theorem for the joint distribution of the central $L$-values corresponding to twists of two distinct primitive Hecke eigenforms. As applications, we obtain (under GRH) positive proportions of twists for which the central $L$-values simultaneously grow or shrink with $q$ as well as a positive proportion of twists for which linear combinations of the central $L$-values are nonzero.