The first power mean of the inversion of \(L\)-functions weighted by quadratic Gauss sums
Let \(L(s,\chi)\) be the Dirichlet \(L\)-function associated with a Dirichlet character \(\chi\) to the modulus~\(q\). The author continues his study of the weighted mean values of the reciprocals of \(L(1,\chi)\). The weights involve the general Gauss sums \[ G(n,\chi;q) = \sum_{a=1}^{q}\chi(a)e^{2\pi i{na^{2}\over q}}. \] The theorems proved in this paper can be stated concisely in the form \[ \sum_{p\leq Q}{1\over p^{m-1}(p-1)^2 }\sum_{\chi \pmod p}{| G(n,\chi;p)| ^{2m}\over | L(1,\chi)| } = A_{m}C\pi(Q) + O(Q^{{1\over 2}}\log^{2}Q), \quad m=1,2. \] Here \(p\) runs over the odd primes up to \(Q\geq 3\), \(n\) is any fixed integer with \(| n| \leq Q\), \(\pi(Q)\) is the number of primes up to \(Q\), \(A_1 = 1,\, A_2 = 3\) and \(C\) is a certain constant. A similar formula is obtained also for the case \(m=3\), with the changes that \(p\) runs up to \(Q\) over the \(\pi_{1}(Q)\) primes \(\equiv 3 \pmod 4\), on the right-hand side \(\pi(Q)\) is replaced by \(\pi_{1}(Q)\), and \(A_3 = 10\). Previously the author and \textit{Y. Deng} [Nagoya Math. J. 167, 1--15 (2002; Zbl 1036.11037)] evaluated similar sums which had \(| L(1,\chi)| ^{2k}\), \(k\) being an arbitrary positive integer, in place of \(| L(1,\chi)| \). As in that paper here too, finding asymptotic formulas in the cases with \(m>3\), and \(m=3\) with \(p\) allowed to run over all primes are announced as unsolved problems.
scientific article; zbMATH DE number 2119046
scientific article