International Journal of Mathematics and Mathematical Sciences
Volume 10 (1987), Issue 2, Pages 395-403
doi:10.1155/S0161171287000462
Functional equation of a special Dirichlet series
Department of Mathematics, Islamic University Gaza, Gaza- Strip, Palestinian Authority
Received 10 December 1985; Revised 1 May 1986
Copyright © 1987 Ibrahim A. Abou-Tair. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
In this paper we study the special Dirichlet series
L(s)=23∑n=1∞sin(2πn3)n−s, s∈C
This series converges uniformly in the half-plane Re(s)>1 and thus represents a holomorphic function there. We show that the function L can be extended to a holomorphic function in the whole complex-plane. The values of the function L at the points 0,±1,−2,±3,−4,±5,… are obtained. The values at the positive integers 1,3,5,… are determined by means of a functional equation satisfied by L.