Vol. 11, No. 9, 2017

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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
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