Exceptional characters and nonvanishing of Dirichlet $L$-functions

Abstract

Let $\psi$ be a real primitive character modulo $D$. If the $L$-function $L(s,\psi )$ has a real zero close to $s=1$, known as a Landau-Siegel zero, then we say the character $\psi$ is exceptional. Under the hypothesis that such exceptional characters exist, we prove that at least fifty percent of the central values $L(1/2,\chi )$ of the Dirichlet $L$-functions $L(s,\chi )$ are nonzero, where $\chi$ ranges over primitive characters modulo $q$ and $q$ is a large prime of size $D^{O(1)}$. Under the same hypothesis we also show that, for almost all $\chi$, the function $L(s,\chi )$ has at most a simple zero at $s=1/2$.

37 pages. To appear in Mathematische Annalen