On the One-Level Density Conjecture for Quadratic Dirichlet L-Functions

Abstract

AbstractIn a previous article, we studied the distribution of “low–lying” zeros of the family of quadratic Dirichlet L–functions assuming the Generalized Riemann Hypothesis for all Dirichlet L–functions. Even with this very strong assumption, we were limited to using weight functions whose Fourier transforms are supported in the interval (–2, 2). However, it is widely believed that this restriction may be removed, and this leads to what has become known as the One-Level Density Conjecture for the zeros of this family of quadratic L-functions. In this note, wemake use of Weil's explicit formula as modified by Besenfelder to prove an analogue of this conjecture.