Dirichlet series related to the Riemann zeta function (Q793774)

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 \(T(s,z)=\sum^{\infty}_{n=1}n^{-s}\sum^{n}_{m=1}m^{- z}(m+n)^{-1}\).