Zeros of quadratic Dirichlet $L$-functions

Abstract

We consider Dirichlet $L$-functions $L(s,\chi )$ where $\chi$ is a quadratic character to modulus $q$. We show that for $\chi$ even (resp. odd) $L(\sigma ,\chi )\ne 0$ for all $\sigma >0$ provided that $q\le 10^9$ (resp. $q\le 10^{10})$. In particular, this result rules out Siegel--Landau zeros for such $L$-functions. Combined, these results improve on previous work by nearly four orders of magnitude. We also list some applications of this work.

arXiv comment12 pages