Weighted central limit theorems for central values of -functions

  • Hung M. Bui

    University of Manchester, UK
  • Natalie Evans

    King’s College London, UK
  • Stephen Lester

    King’s College London, UK
  • Kyle Pratt

    Brigham Young University, Provo, USA
Weighted central limit theorems for central values of $L$-functions cover

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Abstract

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

Cite this article

Hung M. Bui, Natalie Evans, Stephen Lester, Kyle Pratt, Weighted central limit theorems for central values of -functions. J. Eur. Math. Soc. (2024), published online first

DOI 10.4171/JEMS/1417