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Dirichlet series related to the Riemann zeta function. (English) Zbl 0539.10032
For each fixed $$z\in\mathbb C$$ it has been shown that $$H(s,z)$$ defined by the analytic continuation of the Dirichlet series $$H(s,z)=\sum^{\infty}_{n=1}n^{-s}\sum^{n}_{m=1}m^{-z}$$ $$(s,z\in\mathbb C)$$ is a meromorphic function of $$s$$ with first order poles at $$s=1$$, $$s=2-z$$, $$s=1-z$$ and $$s=2-2r-z$$ $$(r\in\mathbb N)$$. (For $$z=1$$ the pole at $$s=1$$ is of second order.) Also for each fixed $$s\neq 1$$ it is shown that $$H(s,z)$$ is a meromorphic function of $$z$$ with first order poles at $$z=1-s$$, $$z=2-s$$ and $$z=2-2r-s$$ $$(r\in\mathbb N)$$. In each case the corresponding residues are determined. Two different representations of $$H(s,z)$$ lead to a reciprocity law $$H(s,z)+H(z,s)=\zeta(s)\zeta(z)+\zeta(s+z)$$ where $$\zeta$$ denotes the Riemann zeta-function. The function values $$H(s,-q)$$ and $$H(-q,z)$$ $$(q\in\mathbb N)$$ are expressed in terms of the Riemann zeta-function. Similar results are obtained for the Dirichlet series $$T(s,z)=\sum^{\infty}_{n=1}n^{-s}\sum^{n}_{m=1}m^{- z}(m+n)^{-1}$$.
Reviewer: Dieter Leitmann

##### MSC:
 11M06 $$\zeta (s)$$ and $$L(s, \chi)$$ 30B40 Analytic continuation of functions of one complex variable 30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
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##### References:
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