An application of Mellin-Barnes type integrals to the mean square of Lerch zeta-functions. II
Let \(s\) be a complex variable, and \(x\) and \(\lambda\) real parameters with \(x>0\). The Lerch zeta-function \(\phi(s,x,\lambda)\) is defined by the meromorphic continuation of the series \(\sum_{n=0}^{\infty}e^{2\pi i\lambda}(n+x)^{-s}\) \((\Re\, s>1)\) over the whole \(s\)-plane; this reduces to the Hurwitz zeta-function \(\zeta(s,x)\) if \(\lambda\) is an integer. Let \(u\) and \(v\) be independent complex variables. \textit{M.~Mikolás} [Acta Sci. Math. Szeged 17, 143--164 (1956; Zbl 0073.06403)] proved an explicit formula for the mean value \(\int_0^1\zeta(u,x)\zeta(v,x)\,dx\) if \(\max(\Re\, u,\Re\, v,\Re(u+v))<1\); otherwise the integral diverges since \(\zeta(s,x)\) has a singularity at \(x=0\). It is hence natural to consider \(\zeta(s,1+x)=\zeta(s,x)-x^{-s}\), for which the singularity in \(x\) is removed. The mean square \(I(s)=\int_0^1|\zeta(s,1+x)|^2\,dx\) was studied little earlier by \textit{J.~F.~Koksma} and \textit{C.~G.~Lekkerkerker} [Indag. Math. 14, 446--452 (1952; Zbl 0047.31501)], who proved that \(I(s)=O\{\log(\Im\, s)\}\) for \(\Im\, s\geq 2\). Improvements upon this result were made by several authors. As for asymptotic aspects of Lerch zeta-functions, the mean square \(I(s;\lambda)=\int_0^1|\phi(s,1+x,\lambda)|^2\,dx\) was first studied by \textit{W.~Zhang} [Adv. Math., Beijing 22, 367--369 (1993; Zbl 0789.11050)], who proved an asymptotic formula for \(I(s;\lambda)\) with the error term \(O\{(\Im\, s)^{-1}\}\) as \(\Im\, s\to+\infty\). The author [Collect. Math. 48, 137--153 (1997; Zbl 0891.11042)] derived a complete asymptotic expansion of \(I(s;\lambda)\) in the descending order of \(\Im\, s\) as \(\Im\, s\to\pm\infty\). The paper under review proceeds with this previous result upon showing that a similar asymptotic series still exists for the multiple mean square \[ \int_0^1\dots\int_0^1|\phi(s,a+x_1+\dots+x_m,\lambda)|^2\,dx_1\dots dx_m \] for any integer \(m\geq1\) and any real \(a>0\), by means of \textit{F.~V.~Atkinson}'s [Acta Math., Uppsala 81, 353--376 (1949; Zbl 0036.18603)] dissection argument applied to the product \(\phi(u,a+x,\lambda)\phi(v,a+x,-\lambda)\). The method of the present paper is also applicable to treat its discrete analogue and their hybridization. In the proofs, Atkinson's method above is enhanced by Mellin-Barnes type integrals, which further enable the author to systematically use various properties of hypergeometric functions.
scientific article; zbMATH DE number 2149100