Über eine Klasse modifizierter Zetafunktionen (Q801102)

If one substitutes in the Euler product expansion \(\zeta (s)=\prod (1- p_ n^{-s})^{-1}\) of the Riemann zeta-function the sequence \((p_ n)\) of the primes by an increasing sequence \((q_ n)\) tending to infinity, then one gets modified zeta-functions \(\zeta^*\). Under certain conditions on the function \(Q(x)=card\{n\in {\mathbb{N}}: q_ n\leq x\}\) the author examines problems concerning the analytic continuation of \(\zeta^*\) and proves some relations between the zeros and singularities of \(\zeta^*\) and those of \(\zeta\). The paper under review generalizes a result of \textit{H. Müller} [Arch. Math. 36, 157-161 (1981; Zbl 0436.10022)].