Two mean value formulae of the Hurwitz zeta-function. (Q2744418)

Let \(0<\alpha\leq 1,\) \(\zeta(s,\alpha)\) be the Hurwitz zeta-function, and \(\zeta'(s,\alpha)\) and \(\zeta''(s,\alpha)\) be its derivatives with respect to \(s\). Define \(\zeta_1(s,\alpha)=\zeta''(s,\alpha)+\alpha^{-s}\log \alpha\) and \(\zeta_2(s,\alpha)=\zeta'(s,\alpha)-\alpha^{-s}.\) Let \(0<\delta_1, \delta_2<1\) and \(\delta_1+\delta_2\not=1.\) Asymptotic formulae are proved for the integrals \(\int_0^1 |\zeta(\delta_1+it,\alpha)\zeta_1(\delta_2-it,\alpha)|\) and \(\int_0^1 |\zeta_2(\delta_1+it,\alpha)\zeta_2(\delta_2-it,\alpha)|.\) The motivation of such research is not recorded.