Exceptional characters and nonvanishing of Dirichlet $L$-functions

TL;DR

This paper investigates the impact of hypothetical exceptional characters with Landau-Siegel zeros on the nonvanishing of central values of Dirichlet L-functions, showing that at least half are nonzero under this assumption.

Contribution

It proves that if exceptional characters exist, then at least fifty percent of the central L-values are nonzero, and most have at most a simple zero at s=1/2.

Findings

At least 50% of L(1/2, χ) are nonzero under the hypothesis.

Almost all L(s, χ) have at most a simple zero at s=1/2.

Results depend on the existence of exceptional characters with Landau-Siegel zeros.

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 $s = 1/2$.