On some conjectures on generalized quadratic Gauss sums and related problems

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 methods and algebraic geometric methods. We prove two conjectures which were proposed recently by the above authors in a previous article (2022). Here we obtain an asymptotic formula for arbitrary power means of generalized quadratic Gauss sums and one corresponding power moment of a character sum.