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Dirichlet series related to the Riemann zeta function. (English) Zbl 0539.10032
For each fixed z it has been shown that H(s,z) defined by the analytic continuation of the Dirichlet series H(s,z)= n=1 n -s m=1 n m -z (s,z) 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). (For z=1 the pole at s=1 is of second order.) Also for each fixed s1 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 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)=ζ(s)ζ(z)+ζ(s+z) where ζ denotes the Riemann zeta-function. The function values H(s,-q) and H(-q,z) (q) are expressed in terms of the Riemann zeta-function. Similar results are obtained for the Dirichlet series T(s,z)= n=1 n -s m=1 n m -z (m+n) -1 .
Reviewer: Dieter Leitmann

11M06ζ(s) and L(s,χ)
30B40Analytic continuation (one complex variable)
30D05Functional equations in the complex domain, iteration and composition of analytic functions