Extended Riemann zeta-functions (Q1414919)

In this paper two extensions of the Riemann zeta-function are presented. Denoting the real part of the complex variable \(s\) by \(\sigma\), these extensions are \[ \zeta_{b}(s) = {1\over \Gamma(s)}\int_{0}^{\infty}t^{s-1}(e^{t}-1)^{-1}e^{-b/t} dt \quad \quad (b>0; b=0, \sigma >1), \] and \[ \zeta_{b}^{*}(s) = {1\over \Gamma(s)(1-2^{1-s})}\int_{0}^{\infty}t^{s-1}(e^{t}+1)^{-1}e^{-b/t} dt \quad \quad (b>0; b=0, \sigma >0). \] Both extensions reduce to the Riemann zeta-function with \(b=0\). As \( b\to 0^{+}\), \(\zeta_{b}(s) \to \zeta(s)\) for \(\sigma >1\), and \(\zeta_{b}^{*}(s)\to \zeta(s)\) for \(\sigma > 0\). Some properties of the extensions are proved. These extension procedures are also applied to the Hurwitz zeta-function.