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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 \(\zeta (s,a)\), and form two parts \(\zeta _{+}\) and \(\zeta _{ - }\) by symmetric and antisymmetric combinations of \(\zeta (s,a)\) and \(\zeta (s,1 - a)\). We consider the properties of \(\zeta _{+}\) and \(\zeta _{ - }\), 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.

11M35 Hurwitz and Lerch zeta functions
Full Text: DOI
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