Fourth power mean of the general $s$-dimensional Kloosterman sum mod $p$

TL;DR

This paper establishes an asymptotic formula for the fourth power mean of general s-dimensional hyper-Kloosterman sums modulo p, advancing understanding of their distribution and behavior.

Contribution

It introduces a new asymptotic formula for the fourth power mean of s-dimensional hyper-Kloosterman sums and analyzes related congruence solutions.

Findings

Derived an asymptotic formula for the fourth power mean

Counted solutions to specific congruence equations mod p

Applied character sum estimates and analytic methods

Abstract

In this article, we prove an asymptotic formula for the fourth power mean of a general $s$-dimensional hyper-Kloosterman sum. We find the number of solutions of certain congruence equations mod $p$ which play an integral part to prove our main result. We use estimates for character sums and analytic methods to prove our theorem.