On some conjectures on Generalized quadratic Gauss sums and related problems

TL;DR

This paper investigates higher power mean values of generalized quadratic Gauss sums, employing advanced mathematical techniques to prove two conjectures related to these sums.

Contribution

It introduces new proofs for two conjectures on generalized quadratic Gauss sums using character sum estimates and algebraic geometry methods.

Findings

Proved two conjectures on quadratic Gauss sums

Established bounds for higher power mean values

Applied novel analytic and algebraic techniques

Abstract

The main purpose of this article is to study higher power mean values of generalized quadratic Gauss sums using estimates for character sums, analytic method and algebraic geometric methods. In this article, we prove two conjectures which were proposed in \cite{BLZ}.