One metric result about analytic continuation of some Dirichlet series

2007/12/10 by Irina Sergeevna Rezvyakova, Irina Rezvyakova, Rezvyakova, Irina
Mathematics · #Fixed Point Theorems Analysis #Meromorphic and Entire Functions #advanced mathematical theories #math.CV #math.NT #msc:11M41

paper · pdf · doi:10.48550/arxiv.0712.1414

4 pages

arxiv created 2007/12/10 · arxiv updated 2009/12/01

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.

Related

• Convergence of families of Dirichlet series
• Analytic continuation of Dirichlet series with almost periodic coefficients
• The Geometry of the Mappings by General Dirichlet Series
• Generalized coefficients of the Dirichlet series
• The zeta function of a recurrence sequence of arbitrary degree