An application of Mellin-Barnes' type integrals to the mean square of Lerch zeta-functions (Q1363142)

Let \(\alpha\) and \(\lambda\) be real parameters with \(\alpha>0\). The zeta-function \[ \varphi (\lambda, \alpha,s)= \sum^\infty_{n=0} e^{2\pi in\lambda} (n+\alpha)^{-s} (\text{Re} s>1) \] was first introduced and studied by \textit{M. Lerch} [Acta Math. 11, 19-24 (1887; JFM 19.0438.01)] and \textit{R. Lipschitz} [J. Reine Angew. Math. 105, 127-156 (1889; JFM 21.0176.01)]. For \(\lambda\in \mathbb{R}\setminus \mathbb{Z}\) it is continued to an entire function over the \(s\)-plane, while for \(\lambda \in\mathbb{Z}\) it reduces to the Hurwitz zeta-function \(\zeta(s, \alpha)\). Define \(\varphi_1 (\lambda, \alpha,s)= \varphi(\lambda, \alpha,s) -\alpha^{-s}\) to remove the singularity at \(\alpha=0\). This paper first deals with the mean square \[ I(s; \lambda)= \int^1_0 \bigl|\varphi_1 (\lambda,\alpha,s) \bigr|^2 d\alpha , \] and derives its complete asymptotic expansion in the descending order of \(\text{Im} s\). This formula gives a natural generalization of an earlier result of \textit{K. Matsumoto} and \textit{M. Katsurada} [Math. Scand. 78, 161-177 (1996; Zbl 0871.11055)] on the mean square of \(\zeta (s,\alpha)\). The proof makes use of a dissection argument of \textit{F. Atkinson} [Acta Math. 81, 353-376 (1949; Zbl 0036.18603)] and Mellin-Barnes type integral formulae. One of the main features of the proof is a systematic application of various properties of Gauss' hypergeometric function. Let \(q\) be an arbitrary positive integer. In the final section the mean square \[ J(s;q, \lambda) =\sum^q_{a=1} \left|\varphi \left(\lambda, {a\over q},s \right) \right |^2, \] which can be regarded as a discrete analogue of \(I(s,\lambda)\), is considered and its complete asymptotic expansion in the descending order of \(q\) is also given.