An identity on the $2m$-th power mean value of the generalized Gauss sums

TL;DR

This paper derives an exact formula for the 2m-th power mean value of generalized quadratic Gauss sums, confirming a conjecture and advancing understanding of their properties using combinatorial and analytic techniques.

Contribution

It provides the first exact calculation of the 2m-th power mean value for generalized quadratic Gauss sums, solving a longstanding conjecture.

Findings

Exact formula for the 2m-th power mean value of generalized quadratic Gauss sums

Confirmation of He and Zhang's conjecture

Advancement in analytic and combinatorial methods for Gauss sums

Abstract

In this paper, using combinatorial and analytic methods, we prove an exact calculating formula on the $2m$-th power mean value of the generalized quadratic Gauss sums for $m\geq 2$. This solves a conjecture of He and Zhang [`On the $2k$-th power mean value of the generalized quadratic Gauss sums', Bull. Korean Math. Soc. 48 (2011), No.1, 9-15].