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

TL;DR

This paper derives and evaluates the sum of the Hurwitz-Lerch zeta function over prime numbers, expressing results in terms of trigonometric functions and Catalan's constant, advancing understanding of special functions at primes.

Contribution

It provides a novel derivation and explicit evaluation of the Hurwitz-Lerch zeta function sum over primes, including special cases involving well-known constants.

Findings

Sum expressed in terms of trigonometric functions

Special cases involve Catalan's constant

Explicit formulas for prime sums

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 $K$.