A Four-Order Linear Recurrence Formula Involving the Quartic Gauss Sums and One Kind Two-Term Exponential Sums

Lan Qi,Xingxing Lv,Tianping Zhang

doi:10.1155/2021/5515014

Abstract

The main purpose of this paper is using the analytic method and the properties of the classical Gauss sums to study the computational problem of one kind hybrid power mean involving the quartic Gauss sums and two-term exponential sums and give an interesting four-order linear recurrence formula for it. As an application, we can obtain all values of this kind hybrid power mean with mathematica software.

Highlights

In the number theory textbooks, especially the elementary number theory and the analytic number theory textbooks, there are many contents related to primes, such as famous prime number theorem and Dirichlet’s theorem
Our goal is to obtain a sharp asymptotic formula for Hk(p). It seems that none had studied it yet, at least we have not seen related papers before. e problem is interesting because it may reveal the regularity of the distribution of values related to the quartic Gauss sums and two-term exponential sums
Let p be an odd prime with p ≡ 1 mod 8. en, for any integer k ≥ 1, we have the four-order linear recurrence formula: Hk+4(p) 6pHk+2(p) + 8pαHk+1(p) −􏼐p2 − 4pα2􏼑Hk(p), (7)

Summary

Introduction

In the number theory textbooks, especially the elementary number theory and the analytic number theory textbooks, there are many contents related to primes, such as famous prime number theorem and Dirichlet’s theorem. M. Shen studied the mean value properties of the quartic Gauss sums and two-term exponential sums and proved an interesting recurrence formula for it. Our goal is to obtain a sharp asymptotic formula for Hk(p) About this problem, it seems that none had studied it yet, at least we have not seen related papers before. We will use the analytic methods and the properties of the classical Gauss sums to study this problem and prove an interesting four-order linear recurrence formula for it. Maybe we can use a similar method to study the general case k ≥ 3, but it is hard to get an exact value for the hybrid power mean as in Hk(p), except k 4. Is is definitely a challenge, which is the goal of our further study

Several Lemmas

Findings

Proofs of the Theorems