On the High-Power Mean of the Generalized Gauss Sums and Kloosterman Sums

Xinyu Liu,Wenpeng Zhang

doi:10.3390/math7100907

Abstract

The main aim of this paper is to use the properties of the trigonometric sums and character sums, and the number of the solutions of several symmetry congruence equations to research the computational problem of a certain sixth power mean of the generalized Gauss sums and generalized Kloosterman sums, and to give two exact computational formulae for them.

Highlights

Let q ≥ 3 be a positive integer, m and n be integers
It is well known that this sum plays an extremely essential role in the research of analytic number theory, and plenty of classical problems in analytic number theory are closely related to it
The main objective of this paper is to apply the properties of the trigonometric sums and character sums, and the number of the solutions of several congruence equations to research the computational problem of (1) and (2) for k = 3, and give two exact computational formulae for them

Summary

Introduction

Let q ≥ 3 be a positive integer, m and n be integers. for any positive integers r > s ≥ 1 and Dirichlet character χ mod q, the generalized Gauss sums G (m, n, r, s, χ; q) is defined as q. We mainly take into account the computational problems of the 2k-th power mean of the generalized Gauss sum and generalized Kloosterman sum. The main objective of this paper is to apply the properties of the trigonometric sums and character sums, and the number of the solutions of several congruence equations to research the computational problem of (1) and (2) for k = 3, and give two exact computational formulae for them. For any odd prime p with 3 - ( p − 1), there exists an integer 1 ≤ m ≤ p − 1 and a non-principal character χ mod p such that the following inequality holds, p −1. For any odd prime p, there exists an integer 1 ≤ m ≤ p − 1 and a non-principal character χ mod p and we can get the inequality p −1.

Some Lemmas

Proofs of the Theorems

Findings

Conclusions