A footnote to mean square value of Dirichlet \(L\)-series
The principal result of this paper is that \[ \sum_{\chi\pmod q} | L({\textstyle {1\over 2}}+it,\chi)|^ 2\;\ll\;(q+t^{5/8} q^{- 1/8})\log qt \qquad\text{for }t\geq 3. \] (The statement given in the abstract is incorrect.) A weaker result for \(| L(\sigma+it,\chi)|^ 2\) with \(0<\sigma<{1\over 2}\) is also given. Previously, \textit{T. Meurman} [Acta Arith. 47, 351-370 (1986; Zbl 0561.10019)] had given the bound \((q+(qt)^{1/3})(qt)^ \varepsilon\) for the case \(\sigma={1\over 2}\), so that the present paper replaces the factor \((qt)^ \varepsilon\) by \(\log qt\), for \(t\leq q^{9/5}\). The proof depends on expressing \(L(s,\chi)\) by means of the Hurwitz zeta- function.
scientific article; zbMATH DE number 495980