Zero free regions for Dirichlet series

Christophe Delaunay (1) , Emmanuel Fricain (1) , Elie Mosaki (1) , Olivier Robert (2)

1 ICJ - Institut Camille Jordan
2 LA MUSE - Laboratoire de Mathématiques Unifiées de Saint-Etienne

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.

Mots clés: zeros of $L$-functions; Dirichlet series; Beurling--Nyman criterion; Hardy spaces; zeros of $L$-functions.

Domaines: Théorie des nombres [math.NT]; Analyse fonctionnelle [math.FA]

Christophe Delaunay, Emmanuel Fricain, Elie Mosaki, Olivier Robert. Zero free regions for Dirichlet series. Transactions of the American Mathematical Society, 2013, 365 (6), pp.3227-3253. ⟨hal-00552945⟩