An application of Mellin-Barnes’ type integrals to the mean square of Lerch zeta-functions. (English) Zbl 0891.11042

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 M. Lerch [Acta Math. 11, 19-24 (1887; JFM 19.0438.01)] and 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 K. Matsumoto and 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 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.


11M35 Hurwitz and Lerch zeta functions
11M06 \(\zeta (s)\) and \(L(s, \chi)\)
33C05 Classical hypergeometric functions, \({}_2F_1\)
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