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Symmetrization of the Hurwitz zeta function and Dirichlet L functions. (English) Zbl 1151.11342
Summary: 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.

MSC:
11M35Hurwitz and Lerch zeta functions