An identity on the $2m$-th power mean value of the generalized 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].