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

Abstract

For the function $f(m,p,q,n)$, where $k,s,a$ general complex numbers and $q$ any positive integer, we establish the sum of values of the Hurwitz-Lerch zeta function $\Phi (f(m,p,q,n),k,a)$ taken at prime numbers $n$. Special cases of this sum are evaluated in terms of products of trigonometric functions and Catalan's constant $K$.