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arXiv · 1804.01445

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

Kyle Pratt
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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.

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Kyle Pratt. 2018-04-04. Average non-vanishing of Dirichlet $L$-functions at the central point. https://doi.org/10.2140/ant.2019.13.227

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