On Classical Gauss Sums and Some of Their Properties

Li Chen

doi:10.3390/sym10110625

Abstract

The goal of this paper is to solve the computational problem of one kind rational polynomials of classical Gauss sums, applying the analytic means and the properties of the character sums. Finally, we will calculate a meaningful recursive formula for it.

Highlights

Since this sum appears in numerous classical number theory problems, and it has a close connection with the trigonometric sums, we believe that classical Gauss sums play a crucial part in analytic number theory
More conclusions have been obtained as regards their arithmetic properties. Such as the following results provided by Chen and Zhang [1]: Let p be an odd prime with p ≡ 1 mod 4, λ be any fourth-order character mod p
The goal of this paper is to use the analytic method and the properties of the character sums to solve the computational problem of Uk ( p, χ), and to calculate two recursive formulae, which are listed hereafter: Theorem 1

Summary

Introduction

Since this sum appears in numerous classical number theory problems, and it has a close connection with the trigonometric sums, we believe that classical Gauss sums play a crucial part in analytic number theory. More conclusions have been obtained as regards their arithmetic properties Such as the following results provided by Chen and Zhang [1]: Let p be an odd prime with p ≡ 1 mod 4, λ be any fourth-order character mod p. The goal of this paper is to use the analytic method and the properties of the character sums to solve the computational problem of Uk ( p, χ), and to calculate two recursive formulae, which are listed hereafter: Theorem 1.

Several Lemmas

Proofs of the Theorems

Findings

Conclusions