New Identities Dealing with Gauss Sums

Wenpeng Zhang,Zhuoyu Chen,Abdul Samad

doi:10.3390/sym12091416

Abstract

In this article, we used the elementary methods and the properties of the classical Gauss sums to study the problem of calculating some Gauss sums. In particular, we obtain some interesting calculating formulas for the Gauss sums corresponding to the eight-order and twelve-order characters modulo p, where p be an odd prime with p=8k+1 or p=12k+1.

Highlights

IntroductionFor any integer q > 1 and any Dirichlet character χ modulo q, the famous Gauss sums G (m, χ; q) is defined as follows:
For any integer q > 1 and any Dirichlet character χ modulo q, the famous Gauss sums G (m, χ; q) is defined as follows: q maG (m, χ; q) = ∑ χ( a)e, q a =1 where m is any integer and e(y) = e2πiy .If χ is any primitive character modulo q or m co-prime to q (that is, (m, q) = 1), we have the identityG (m, χ; q) = χ(m) G (1, χ; q) ≡ χ(m)τ (χ).If χ is a primitive character modulo q, for any integer m, we have the following two important identities: q χ(m) = · χ(b) e τ (χ) b∑ =1
W.P. and Hu, J.Y. [1] or Berndt, B.C. and Evans, R.J. [8] studied the properties of Gauss sums of the third-order character modulo p, and proved the following result: Let p be a prime with p ≡ 1 mod 3

Summary

Introduction

For any integer q > 1 and any Dirichlet character χ modulo q, the famous Gauss sums G (m, χ; q) is defined as follows:. [8] studied the properties of Gauss sums of the third-order character modulo p, and proved the following result: Let p be a prime with p ≡ 1 mod 3. [3] 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. [4] studied the properties of the Gauss sums of the sixth-order character modulo p, and deduced an interesting identity (see Lemma 1 below). If p be an odd prime with p ≡ 1 mod 8, for any eighth-order characters χ8 modulo p, we have the identities τ 4 (χ8 ) + τ 4 χ38 = τ 4 (χ8 ) + τ 4 χ38 = 2p 2 · |α|.

Proof of the Theorems

Findings

Conclusions