On the weighted mean value of the inversion of Dirichlet \(L\)-functions (Q2739284)

Define a multiplicative function \(r(n)\) by \(r(p^k)=-((2k)!/(k!)^2)/4^k(2k-1)\) and let \(B(q)=\prod_{p\equiv 0\bmod q} \sum_{k\geq 0}r(p^k)^2p^{-2k}\). Further, denote by \(J(q)\) the number of primitive characters \(\bmod q\). In the present paper the authors study the weighted mean value of the reciprocals of Dirichlet \(L\)-functions \(L(s,\chi)\). They prove the asymptotic formula NEWLINE\[NEWLINE \sum_{q\leq Q}{B(q)\over J(q)}\sum_{\chi\bmod q}{1\over |L(1,\chi)|}={3\over 4}Q\sum_{n=1}^\infty {r(n)^2\over n^2}+O(Q^{1/2}(\log Q)^{5/2}),NEWLINE\]NEWLINE where the summation on the left hand side is taken over all \(q\not\equiv 2\bmod 4\), and all primitive characters \(\bmod q\) for \(q\leq Q, Q>2\). The proof relies mainly on \textit{R. C. Vaughan}'s estimates for Möbius' \(\mu\)-function [Recent progress in analytic number theory, Symp. Durham 1979, Vol. 1, 341-348 (1981; Zbl 0483.10041)].