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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.

MSC:
 11M35 Hurwitz and Lerch zeta functions
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References:
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