A Generalization of the Riemann Functional Equation with Three Variables

Abstract

The discovery that the generalized Riemann functional equation from the literature can be extended to real $b \ge 0$, by exploiting symmetries of a new formula for the Hurwitz zeta function with respect to the shift parameter $(b)$, led to the search for a similar relation for the Lerch $\Phi$ function, $\Phi(e^z,k,b)$, where a new formula is used as a starting point and the symmetries with respect to $b$ are exploited again. This search resulted in a generalization of the Riemann functional equation with three variables, whose domain is then extended to $b>1$, with the aid of a new self-contained Lerch $\Phi$ formula, making the relation hold for all real $b$. As a secondary objective, we address the problem of obtaining the analytic continuation of a finite summation that appears in a formula for the Lerch $\Phi$ function. Above all, this paper demonstrates, with clear elementary mathematics, how a functional equation can be derived.