Average non-vanishing of Dirichlet $L$-functions at the central point

TL;DR

This paper demonstrates that unconditionally, over averages of primitive characters and moduli, at least 50.073% of Dirichlet L-functions at the central point are non-zero, surpassing the GRH implication of 50%.

Contribution

It introduces an unconditional average non-vanishing result for Dirichlet L-functions at the central point, improving upon the GRH-based bound using mollification and Kloosterman sum estimates.

Findings

At least 50.073% of central L-values are non-vanishing on average.

Utilizes mollification with a three-piece mollifier.

Employs estimates for sums of Kloosterman sums.

Abstract

The Generalized Riemann Hypothesis implies that at least 50% of the central values $L \left( \frac{1}{2},\chi\right)$ are non-vanishing as $\chi$ ranges over primitive characters modulo $q$. We show that one may unconditionally go beyond GRH, in the sense that if one averages over primitive characters modulo $q$ and averages $q$ over an interval, then at least 50.073% of the central values are non-vanishing. The proof utilizes the mollification method with a three-piece mollifier, and relies on estimates for sums of Kloosterman sums due to Deshouillers and Iwaniec. Note: The author has been made aware of an error in this work. It seems the error can be fixed, by using a different argument, and the author will present a correction in due course.