Explicit formulas and asymptotic expansions for certain mean square of Hurwitz zeta-functions
Let \(\zeta(s,\alpha)\) (\(0<\alpha\leq 1\)) be the Hurwitz zeta-function and let \(\zeta_ 1 (s,\alpha)= \zeta(s,\alpha)- \alpha^{-s}\). The full asymptotic expansion of \(\int_ 0^ 1 \zeta_ 1(u,\alpha) \zeta_ 1 (v,\alpha)d\alpha\) is stated, where \(u\) and \(v\) are complex numbers satisfying certain natural conditions. From this result several other interesting results are obtained, such as the asymptotic expansion of \(\int_ 0^ 1 |\zeta_ 1 ({1\over 2}+ it, \alpha)|^ 2 d\alpha\), which sharpens previous results. The intricate proofs, which will appear elsewhere, are based on the methods of \textit{F. V. Atkinson} [Acta Math. 81, 353-376 (1949; Zbl 0036.186)], \textit{Y. Motohashi} [Proc. Japan Acad., Ser. A 61, 222-224 (1985; Zbl 0573.10027)] and the authors [Math. Z. 208, 23-39 (1991; Zbl 0744.11041)].
scientific article; zbMATH DE number 549039