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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.

11M35Hurwitz and Lerch zeta functions
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