Asymptotic equalities for the Lipschitz-Lerch zeta-function
The Lipschitz-Lerch zeta-function \(L(x,a,s)=\sum_{n\geq 0}\exp (2\pi inx)(n+a)^{-s}\), \(0<a\leq 1\), Re s\(>1\) has an analytic continuation to the whole complex s-plane if \(x\in {\mathbb{R}}\setminus {\mathbb{Z}}\). For \(x\in {\mathbb{Z}}\) it reduces to the meromorphic Hurwitz zeta-function \(\zeta(s,a)\). The author studies the asymptotic behavior of mean values in the `critical strip' \(0<Re s<1\), esp. on the `critical line' Re s\(=.\) To give an example of his results we mention theorem 1: Let \(0<a\leq 1\), \(x\in {\mathbb{R}}\setminus {\mathbb{Z}}\). As \(\delta\to 0\) \[ \int^{\infty}_{0}| L(x,a,+it)|^ 2 e^{-\delta t} dt\;\sim \;\delta^{-1} \log \delta^{-1}. \] The method of proof is based on the theory of Fourier transforms and on Parseval's formula and does not depend on ``approximate functional equations''.
scientific article; zbMATH DE number 4035931