On analytic continuation and functional equation of certain Dirichlet series

E Carletti,G Monti Bragadin

doi:10.1155/s0161171293000742

Abstract

Analytic continuation and functional equation of Riemann's type are proved for a class of Dirichlet series associated to rational functions.

Highlights

In this paper we are concerned with analytic continuation and functional equation
Our objective is to show how a very classical method, which is one of Rim’ methods, works for a quite general class of Dirichlet series
In the course of the proof it turns out that the Dirichlet series we consider in our paper are expressed in terms of Hurwitz zeta functions

Summary

INTRODUCTION

In this paper we are concerned with analytic continuation and functional equation In the course of the proof it turns out that the Dirichlet series we consider in our paper are expressed in terms of Hurwitz zeta functions. This result should be compared with Theorem of Arakawa [3] where a representation of E cotgamr n in terms of Barnes zeta functions is given. The last series in (2.3) defines a meromorphic function with simple poles at s no and possibly at s n < no, since it converges uniformly on any compact subset of { e C: Is + n[ >_ C,n > no}, C being fixed positive constant. --FIO so that our first claim follows, because of the poles of r(s) at non-positive integers. The second claim follows from the above argument with no 0

FUNCTIONAL EQUATION OF CERTAIN DIRICHLET SERIES

The corresponding Dirichlet series