Higher order moments of generalized quadratic Gauss sums weighted by $L$-functions

TL;DR

This paper derives asymptotic formulas for higher order moments of generalized quadratic Gauss sums weighted by $L$-functions, confirming a conjecture and advancing understanding of their distribution using character sum estimates.

Contribution

It provides the first asymptotic formulas for the 6th and 8th moments of these sums, using novel estimates for related character sums.

Findings

Asymptotic formulas for 6th and 8th moments obtained

Confirmation of Wenpeng Zhang's conjecture

Enhanced understanding of the distribution of quadratic Gauss sums

Abstract

The main purpose of this paper is to study higher order moments of the generalized quadratic Gauss sums weighted by $L$-functions using estimates for character sums and analytic methods. We find asymptotic formulas for three character sums which arise naturally in the study of higher order moments of the generalized quadratic Gauss sums. We then use these character sum estimates to find asymptotic formulas for the $\nth{6}$ and $\nth{8}$ order moments of the generalized quadratic Gauss sums weighted by $L$-functions. Our asymptotic formulas satisfy a conjecture of Wenpeng Zhang.