A New Gauss Sum and Its Recursion Properties

Li Chen,Wenpeng Zhang

doi:10.1155/2021/6165911

Abstract

In this paper, we introduce a new Gauss sum, and then we use the elementary and analytic methods to study its various properties and prove several interesting three-order linear recursion formulae for it.

Highlights

Studied the properties of some special Gauss sums and obtained the following interesting results: let p be a prime with p ≡ 1 mod 3. en for any third-order character λ modulo p, one has the identity τ3(λ) + τ3(λ) dp, (3)
Chen and Zhang [5] studied the case of the fourth-order character modulo p and obtained the following conclusion: let p be a prime with p ≡ 1 mod 4. en for any four-order character χ4 modulo p, we have the identity τ2
We introduce a new Gauss sum A(m) A(m, p), which is defined as follows: let p be an odd prime

Summary

Introduction

For any Dirichlet character χ modulo q, the classical Gauss sums G(m, χ; q) is defined as follows:. Studied the properties of some special Gauss sums and obtained the following interesting results: let p be a prime with p ≡ 1 mod 3. Chen and Zhang [5] studied the case of the fourth-order character modulo p and obtained the following conclusion: let p be a prime with p ≡ 1 mod 4. En for any four-order character χ4 modulo p, we have the identity τ2. It is clear that if (p − 1, 3) 1, note that χ32 χ2; from the properties of the reduced residue system modulo p, we have χ2 This time, A(m) G(m, χ2; p) χ2(m)τ(χ2) becomes the classical Gauss sum. B4 , where b is the same as defined in (3), i.e., 4p d2 + 27b2

Several Lemmas

Findings

Proofs of the Theorems