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Zbl 0539.10032
Apostol, Tom M.; Vu, Thiennu H.
Dirichlet series related to the Riemann zeta function.
(English)
[J] J. Number Theory 19, 85-102 (1984). ISSN 0022-314X; ISSN 1096-1658/e

For each fixed $z\in\Bbb C$ it has been shown that $H(s,z)$ defined by the analytic continuation of the Dirichlet series $H(s,z)=\sum\sp{\infty}\sb{n=1}n\sp{-s}\sum\sp{n}\sb{m=1}m\sp{-z}$ $(s,z\in\Bbb C)$ is a meromorphic function of $s$ with first order poles at $s=1$, $s=2-z$, $s=1-z$ and $s=2-2r-z$ $(r\in\Bbb N)$. (For $z=1$ the pole at $s=1$ is of second order.) Also for each fixed $s\ne 1$ 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 $z=2-2r-s$ $(r\in\Bbb N)$. 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 $H(-q,z)$ $(q\in\Bbb N)$ are expressed in terms of the Riemann zeta-function. Similar results are obtained for the Dirichlet series $T(s,z)=\sum\sp{\infty}\sb{n=1}n\sp{-s}\sum\sp{n}\sb{m=1}m\sp{- z}(m+n)\sp{-1}$.
[Dieter Leitmann]
MSC 2000:
*11M06 Riemannian zeta-function and Dirichlet L-function
30B40 Analytic continuation (one complex variable)
30D05 Functional equations in the complex domain

Keywords: analytic continuation; Dirichlet series; meromorphic function; first order poles; residues; reciprocity law; Riemann zeta-function

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