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

We establish a central limit theorem for the central values of Dirichlet Lfunctions with respect to a weighted measure on the set of primitive characters modulo q as q → ∞. 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) a positive proportion of twists for which the central L-values simultaneously grow or shrink with q as well as a positive proportion of twists for which the central L-values are distinct and simultaneously non-zero.

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