On the density of zeros of linear combinations of Euler products for σ > 1

Righetti, Mattia

doi:10.2140/ant.2017.11.2131

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On the density of zeros of linear combinations of Euler products for $\sigma \gt {}$1

Mattia Righetti

Vol. 11 (2017), No. 9, 2131–2163

DOI: 10.2140/ant.2017.11.2131

Abstract

It has been conjectured by Bombieri and Ghosh that the real parts of the zeros of a linear combination of two or more $L$ -functions should be dense in the interval $[1, σ^{*}]$ , where $σ^{*}$ is the least upper bound of the real parts of such zeros. In this paper we show that this is not true in general. Moreover, we describe the optimal configuration of the zeros of linear combinations of orthogonal Euler products by showing that the real parts of such zeros are dense in subintervals of $[1, σ^{*}]$ whenever $σ^{*} > 1$ .

Keywords

zeros of Dirichlet series, value distribution, asymptotic distribution functions, convexity

Mathematical Subject Classification 2010

Primary: 11M41

Secondary: 11M26

Milestones

Received: 21 November 2016

Revised: 18 July 2017

Accepted: 14 August 2017

Published: 2 December 2017

Authors

Mattia Righetti
Dipartimento di Matematica Università di Genova 16146, Genova Italy	Centre de Recherches Mathématiques Université de Montréal P.O. Box 6128 Centre-Ville Station Montréal QC H3C 3J7 Canada

Vol. 11, No. 9, 2017