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arXiv · 2109.06829

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

Hung M. Bui·Natalie Evans·Stephen Lester·Kyle Pratt
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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.

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Hung M. Bui, Natalie Evans, Stephen Lester, Kyle Pratt. 2021-09-14. Weighted central limit theorems for central values of $L$-functions. https://arxiv.org/abs/2109.06829

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