A weighted integral approach to the mean square of Dirichlet \(L\)-functions (Q2707570)

Let \(E(T,\chi)\) be the error term of the mean square formula for the Dirichlet \(L\)-function \(L(s,\chi)\) on the critical line \(\sigma = 1/2\), and let \(E_{\sigma}(T,\chi)\) be the analogous error term for \(1/2<\sigma <1\). The corresponding error terms \(E(T)\) and \(E_{\sigma }(T)\) related to Riemann's zeta-function have been extensively studied, and the object of the present paper is to generalize some results to \(L\)-functions. It is assumed that \(\chi \) is primitive character \(\pmod{p}\) with \(p\) a prime; the latter condition is needed when Weil's estimate for Kloosterman sums is used, but the authors point out that the case of composite moduli is probably also manageable. The main result is that NEWLINE\[NEWLINEE_{\sigma }(T, \chi) \ll (pT)^{1/(1+4\sigma)}\log pT + p^{1/2}(\log pT)^{1+2\sigma }NEWLINE\]NEWLINE for any fixed \(\sigma \in (1/2, 1)\), and for \(\sigma =1/2\) this gives an estimate for \(E(T,\chi)\) up to minor modifications in the logarithmic factors. The principle of the proof is to start from a ''local'' mean square equipped with a Gaussian weight, and to use this to deduce an approximation for the ''global'' mean square; at the first stage, a generalization of Atkinson's dissection method is used.NEWLINENEWLINEFor the entire collection see [Zbl 0932.00040].