On the Hybrid Fourth Power Mean Involving Legendre’s Symbol and One Kind Two-Term Exponential Sums

Jingzhe Wang,Tingting Wang

doi:10.1155/2021/5560902

Jingzhe Wang, Tingting Wang

Open Access

https://doi.org/10.1155/2021/5560902

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Abstract

The main purpose of this article is using the elementary methods and the properties of the quadratic residue modulo an odd prime p to study the calculating problem of the fourth power mean of one kind two-term exponential sums and give an interesting calculating formula for it.

Highlights

A 0 q where, as usual, e(y) e2πiy and i denotes the imaginary unit, that is i2 − 1. Since this kind of sums play a very important role in the study of analytic number theory, so many number theorists and scholars had studied the various properties of G(m, k; q) and obtained a series of meaningful research results, we do not want to enumerate here, and interested readers can refer to [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]
Where p is an odd prime and h ≥ 2 is an integer. It seems that none had studied it before; at least we have not seen such a result at present
Let p be an odd prime with p ≡ 1mod4. en, for any integer h ≥ 2, there are two integers C C(h, p) and D D(h, p) depending only on h and p, such that the identity p− 1

Summary

Introduction

Chen and Zhang [3] proved that for any prime p with p ≡ 5 mod 8, one has the identity p− 1. Chen and Wang [5] studied the calculating problem of the fourth power mean of G(m, 4; p) and proved the following conclusion. Inspired by the works in [5, 6], in this paper, we consider the following calculating problem of the 2h-th power mean of the two-term exponential sums: p− 1. Where p is an odd prime and h ≥ 2 is an integer About this problem, it seems that none had studied it before; at least we have not seen such a result at present. Let p > 3 be an odd prime with p ≡ 1mod, we have the asymptotic formula p− 1.

Several Lemmas

Proof of the Theorem