A Note on the Classical Gauss Sums

Tingting Wang,Guohui Chen

doi:10.3390/math6120313

Tingting Wang, Guohui Chen

Open Access

https://doi.org/10.3390/math6120313

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Abstract

The main purpose of this paper is to study the computational problem of one kind rational polynomials of the classical Gauss sums, and using the purely algebraic methods and the properties of the character sums mod p ( a prime with p ≡ 1 mod 12 ) to give an exact evaluation formula for it.

Highlights

As usual, let q ≥ 3 be an integer, χ be any Dirichlet character mod q
Many number theory experts have studied the properties of the classical Gauss sums, and obtained a series of important conclusions
Zhang [1] provided the following result: Let p be an odd prime with p ≡ 1 mod 4, λ be any fourth-order character mod p

Summary

Introduction

Let q ≥ 3 be an integer, χ be any Dirichlet character mod q. This sum occupies a very vital position in the research of analytic number theory, and plenty of famous number theory problems are closely related to it Because of this reason, many number theory experts have studied the properties of the classical Gauss sums, and obtained a series of important conclusions. P. Zhang [1] provided the following result: Let p be an odd prime with p ≡ 1 mod 4, λ be any fourth-order character mod p. Y. Hu [2] used identity (1) to obtain a second-order linear recursive formula for one kind rational polynomials involving the classical Gauss sums. For any prime p with p ≡ 1 mod 12 and any third-order character ψ mod p, one has the second-order linear recursive formula. Let p be a prime with p ≡ 1 mod 12, χ be any twelfth-order character mod p.

Two Simple Lemmas

Proof of the Theorem

Conclusions