Moments of generalized quadratic Gauss sums weighted by \(L\)-functions (Q5960998)

The author studies moments of the generalized quadratic Gauss sum NEWLINE\[NEWLINEG(n, \chi; q)=\sum_{a=1}^q\chi(a)\exp(2\pi ina^2/q),\;q,n\in\mathbb Z,\;q\geq 2,NEWLINE\]NEWLINE weighted by the Dirichlet \(L\)-function \(L(s, \chi)\), where \(\chi\) is the character modulo \(q\). Let \(p\) denote an odd prime and let \((n, p)=1\). Using estimates for character sums and the analytic methods, he establishes the following asymptotic formulas: NEWLINE\[NEWLINE\sum_{\chi\neq\chi_0}|G(n, \chi; p)|^{2k} |L(1, \chi)|=\begin{cases} Cp^2+O(p^{3/2}\log^2 p), &\text{if }k=1,\\ 3Cp^3+O(p^{5/2}\log^2 p), &\text{if }k=2,\\ 10Cp^4+O(p^{7/2}\log^2 p), &\text{if }k=3 \text{ and }p\equiv 3\bmod 4,\end{cases} NEWLINE\]NEWLINE where \(C\) is the absolute constant explicitly given in the paper \(\sum_{\chi\neq\chi_0}\) denotes the summation over all nonprincipal characters modulo \(p\). Then the author gives explicit formulas for sums \(\sum_{\chi \bmod p}|G(n, \chi; p)|^{2k}\), if \(k=2\), \(p\equiv 1,3\bmod 4\), also if \(k=3\), \(p\equiv 3\bmod 4\). Based on the results obtained he raises the conjecture, that for any positive integer \(k\), NEWLINE\[NEWLINE\sum_{\chi\neq\chi_0}|G(n, \chi; p)|^{2k} |L(1, \chi)|\sim C\sum_{\chi \bmod p}|G(n, \chi; p)|^{2k}, \quad p\to +\infty.NEWLINE\]