On the hybrid mean value of Gauss sums and generalized Bernoulli numbers (Q1764340)

For positive integers \(n\), \(m\) and \(q\geq 3\) the authors consider sums \(\sum_\chi \tau^m(\overline\chi) B^m_{n,\chi}\) extended over all nonprincipal Dirichlet characters \(\chi\text{\,mod\,}q\), where \(\tau(\chi)\) is a Gauss sum associated with \(\chi\) and \(B_{n,\chi}\) is a generalized Bernoulli number. They show that these sums satisfy an asymptotic formula of the type \(F_{m,n}(q)+ O(q^{nm+\varepsilon})\), where \(F_{m,n}(q)\) is given explicitly in closed form.