Sum of the Hurwitz-Lerch Zeta Function over Natural Numbers: Derivation and Evaluation

Abstract

We consider a Hurwitz-Lerch zeta function Φs,z,a sum over the natural numbers. We provide an analytically continued closed form solution for this sum in terms of the addition of Hurwitz-Lerch zeta functions. A new recurrence identity with consecutive neighbours and the product of trigonometric functions is derived.