On the One-Level Density Conjecture for Quadratic Dirichlet L-Functions
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.
11M26 - Nonreal zeros of $\zeta (s)$ and $L(s, \chi)$; Riemann and other hypotheses