A hybrid mean value involving Gauss sums and character sums (Q1959032)

Let \(q=MN>2\) be an integer , \(M=\prod_{p\parallel q}p,(M,N)=1.\) The authors prove that for any real number \(Q\) and any fixed positive number \(\epsilon \) with \(1<Q<q^{1-\varepsilon } \), we have the asymptotic formula \[ \sum_{{\chi\bmod q}\atop {\chi \neq \chi_0}}|\tau(\chi)|^m\cdot\biggl|\sum_{n\leq Q}\chi(n)\biggr|^2\cdot \biggl|\frac{L'}{L}(1,\chi)\biggr|^2=\frac{\varphi(q)\phi^2(N)}{q}QN^{\frac{m}{2}-1}C\cdot \] \[ \prod_{p|M}\left(p^{\frac{m}{2}+1}-2p^{\frac{m}{2}}+1\right)+O\left(2^{\omega(q)}q^{\frac{m}{2} +\varepsilon}\right)+O\left(Q^2q^{\frac{m}{2}+\varepsilon}\right), \] where \(\sum_{\chi\bmod q}\atop {\chi \neq \chi_0} \) denotes the summation over all non-principal characters modulo \(q\), \( \varphi(q)\) is the Euler function, \(\omega(q)\) the number of the different prime divisors of \(q\) and \(C\) is a constant depending only of \(q\).