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

http://dx.doi.org/10.4153/CJM-2006-034-9
Canad. J. Math. 58(2006), 843-858

  Published:2006-08-01
 Printed: Aug 2006

In a previous article, we studied the distribution of ``low-lying" zeros of the family of quad\-ratic 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, we make use of Weil's explicit formula as modified by Besenfelder to prove an analogue of this conjecture.