## 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.

### MSC:

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

### Citations:

Zbl 0871.11055; Zbl 0036.18603; JFM 19.0438.01; JFM 21.0176.01
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