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


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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