Symmetrization of the Hurwitz zeta function and Dirichlet L functions

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Journal Article
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2007, 463 (2077), pp. 281 - 301
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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. © 2006 The Royal Society.
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