Discrete mean values of Hurwitz zeta-functions
Let \(\zeta (s,a) = \sum^ \infty_{n=0} (n+a)^{-s}\) be the Hurwitz zeta-function, and set \[ I(s,q) = \sum^ q_{a=1} \bigl | \zeta (s,a/q) \bigr |^ 2. \] This sum is of relevence to questions involving the mean-square of Dirichlet \(L\)-functions \(L(s,\chi)\) averaged over \(\chi \pmod q\). \textit{W. Zhang} [Acta Math. Sin. New Ser. 7, 103-118 (1991; Zbl 0732.11040)] gave an asymptotic formula for \(I({1 \over 2}+it,q)\) when \(t \geq q^{6/5 - \delta}\). The present paper gives a full asymptotic series when \(t \leq q^{1/2 - \delta}\). This is of the form \[ I ({\textstyle{1 \over 2}}, q) = q \log (q/2 \pi) + 2 \gamma + \text{Re} \Gamma^ q/ \Gamma ({\textstyle {1 \over 2}} + it) + \sum^ N_{n=0} \text{Re} \bigl( q^{1/2 + it - n} f_ n(t) \bigr) + R_ N(q,t), \] with explicit functions \(f_ n(t)\) and \(R_ N (q,t)\), for which \(R_ N (q,t) \ll_ N t^{2N} q^{1/2 - N}\) \((t \gg 1)\). This is deduced from a corresponding result with \(0<\sigma<1\), \(\sigma \neq 1/2\). The proof is a variant of the argument used in the authors' earlier work [Math. Z. 208, 23-39 (1991; Zbl 0744.11041)] on the mean square of Dirichlet \(L\)-functions.
mean-square of Dirichlet \(L\)-functions
scientific article; zbMATH DE number 556444
scientific article