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

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 non-zero.