On the \(2k\)-th power mean of inversion of \(L\)-functions with the weight of the Gauss sum
The following result is proved. Let \(k\) be a fixed positive integer, \(Q >1\) and \(m > 0\). One has \[ \sum_{q\leq Q}A(q,k)\sum_{\chi (\bmod q)}{| \tau(\chi)| ^{m}\over | L(1,\chi)| ^{2k}} = {2\over m+4}Q^{{m\over 2}+2} + O(Q^{{m\over 2}+1+\epsilon}). \] Here \(\chi\) runs through the Dirichlet characters to the modulus \(q = NM\) with \(M=\prod_{p| | q}p\), and \[ A^{-1}(q,k) = \prod_{p| N}(1-{1\over p})^{2}\prod_{p| M}(1-{2\over p}+{1\over p^{{m\over 2}+1}})\prod_{p\nmid q}(1+{{k\choose 1}^{2}\over p^{2}}+\cdots +{{k\choose k}^{2}\over p^{2k}}). \]
scientific article; zbMATH DE number 2119283