One metric result about analytic continuation of some Dirichlet series

TL;DR

This paper proves that most series in a specific family of Dirichlet series can be analytically continued to Re s > 1/2 without zeros, using estimates of trigonometric sums, providing a simpler proof of a known result.

Contribution

It offers a simplified proof of the analytic continuation and non-vanishing of almost all series in a parametric family of Dirichlet series, extending previous results.

Findings

Almost all series in the family have analytic continuation to Re s > 1/2.

These series do not vanish in the half-plane Re s > 1/2.

The proof uses estimates of trigonometric sums.

Abstract

In this paper we consider certain 1-parametric family of Dirichlet series. For a particular value of the parameter the series turns into the Dirichlet series for the Riemann zeta function. We prove that almost every series of the family has analytic continuation to the half plane Re s > 1/2 where it doesn't vanish. The result was obtained before by different authors. We give its simple proof in terms of estimates of some trigonometric sums.