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

Daniel Fiorilli

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arXiv · 1403.7079

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

Daniel Fiorilli

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Abstract

We investigate the consequences of natural conjectures of Montgomery type on the non-vanishing of Dirichlet $L$-functions at the central point. We first justify these conjectures using probabilistic arguments. We then show using a result of Bombieri, Friedlander and Iwaniec and a result of the author that they imply that almost all Dirichlet $L$-functions do not vanish at the central point. We also deduce a quantitative upper bound for the proportion of Dirichlet $L$-functions for which $L(\frac 12,χ)=0$.

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Daniel Fiorilli. 2014-03-27. On the non-vanishing of Dirichlet $L$-functions at the central point. https://arxiv.org/abs/1403.7079

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