Symmetrization of the Hurwitz zeta function and Dirichlet L functions

Abstract

We consider the Hurwitz zeta function ζ(s,a), and form two parts ζ+ and ζ- by symmetric and antisymmetric combinations of ζ(s,a) and ζ(s,1-a). We consider the properties of ζ+ and ζ-, and then show that each may be decomposed into parts denoted by P and N, each of which obeys a functional equation of the Dirichlet L type, with a multiplicative factor of -1 for the functions N. We show the results of this procedure for rational a=p/q, with q=1, 2, 3, 4, 5, 6, 7, 8, 10, and demonstrate that the functions P and N have some of the key properties of Dirichlet L functions. Show more