Zero free regions for Dirichlet series

TL;DR

This paper investigates explicit zero-free regions for Dirichlet series and extends Beurling-Nyman criteria to $L$-functions, with applications to the Siegel zero problem, improving upon previous results by Nikolski and de Roton.

Contribution

It introduces generalized zero-free regions and a broader Beurling-Nyman criterion applicable to $L$-functions, advancing the understanding of zero distributions.

Findings

Established explicit zero-free discs for certain Dirichlet series.

Generalized Beurling-Nyman criterion for $L$-functions.

Applied results to the Siegel zero problem.

Abstract

In this paper, we are interested in explicit zero-free discs for some Dirichlet series and we also study a general Beurling-Nyman criterion for $L$-functions. Our results generalize and improve previous results obtained by N. Nikolski and by A. de Roton. As a concrete application, we get, for example, a Beurling-Nyman type criterion for the Siegel zero problem.