On the classical Gauss sum and the recursive properties

Hui Bai,Jiayuan Hu

doi:10.1186/s13662-018-1804-7

Abstract

Let p be a prime with pequiv1bmod8, ψ be an eighth character modp, and tau(psi) denote the classical Gauss sum modp. The main purpose of this paper is using the analytic method and the properties of the classical Gauss sum to study the computational problem of one kind rational polynomial of tau(psi). In the end, we prove an interesting second-order linear recursive formula for it.

Highlights

From some special characters χ mod q, the Gauss sum τ (χ) has some interesting properties
For any positive integer q ≥ 2 and Dirichlet character χ mod q, the classical Gauss sum τ (χ) is defined as q a τ (χ) = χ(a)e, q a=1 where e(y) = e2πiy.This sum is very important in the study of the analytic number theory, so many authors had studied its elementary properties, and obtained a series of important results
Theorem 1 Let p be a prime with p ≡ 1 mod 8, for any integer k ≥ 2, we have the second-order linear recursive formula

Summary

Introduction

From some special characters χ mod q, the Gauss sum τ (χ) has some interesting properties. If p is a prime with p ≡ 1 mod 3, and ψ is any third-order character mod p, we have the identity [1, 2] We are considering such a sequence Fk(p) as follows: Let p be a prime with p ≡ 1 mod 8, ψ be any eighth-order character modp.

Objectives

Results

Conclusion