A New Generalization of the Riemann Functional Equation

Abstract

A new integral representation for the Hurwitz zeta function, $\zeta(s,b)$, can be manipulated in a way as to make the integral part disappear from the formula, leading to a new relation between the Hurwitz zeta and the polylogarithm that holds for all complex $s \ne 1$ and positive $b$. This is achieved through the symmetries between $\zeta(s,b)$ and $\zeta(s, -b)$.