Moments of Kummer sums weighted by \(L\)-functions (Q6596352)

Exponential sums of the form \N\[\NS_p =\sum_{a=1}^pe\left(\frac{a^3}{p}\right),\N\]\Nwhere \(e(y)=e^{2\pi i y}\), were studied by Kummer, see [\textit{E. E. Kummer}, J. Reine Angew. Math. 32, 341--359 (1846; ERAM 032.0930cj)]. The paper under review is devoted to Kummer sums weighted by Dirichlet character \(\chi\): \N\[\NS_p(n ; \chi)=\sum_{a=1}^p \chi(a) e\left(\frac{n a^3}{p}\right).\N\]\NThe main purpose of this article is to study higher order moments of such sums.\N\NTheorem 1.2. Let \(p\) an odd prime such that \(p \equiv 1 \bmod 3\) and \(n\) be an integer. Then, for any Dirichlet character \(\chi \bmod p\), we have an asymptotic formula \N\[\N\sum_{\chi \bmod p}\left|S_p(n ; \chi)\right|^2=(p-1)^2.\N\]\NTheorem 1.3. Let \(p\) an odd prime such that \(p \equiv 1 \bmod 3\) and \(n\) be an integer. Then, for any Dirichlet character \(\chi \bmod p\), we have an asymptotic formula \N\[\N\sum_{\chi \bmod p}\left|S_p(n ; \chi)\right|^4=5 p^3+O\left(p^{5 / 2}\right).\N\]\NLast two theorems give asymptotic formulae for the sums \N\[\N\sum_{\chi \ne \chi_0 }\left|S_p(n ; \chi)\right|^2\left|L(1, \chi)\right| \N\]\Nand \N\[\N\sum_{\chi \ne \chi_0 }\left|S_p(n ; \chi)\right|^4\left|L(1, \chi)\right|.\N\]