A new fourth power mean of two-term exponential sums

Chen Li,Wang Xiao

doi:10.1515/math-2019-0034

Abstract

Abstract The main purpose of this paper is to use analytic methods and properties of quartic Gauss sums to study a special fourth power mean of a two-term exponential sums modp, with p an odd prime, and prove interesting new identities. As an application of our results, we also obtain a sharp asymptotic formula for the fourth power mean.

Highlights

We will use many properties of the classical Gauss sums, the fourth-order character mod p and the quartic Gauss sums. All of these contents can be found in any Elementary Number Theory or Analytic Number Theory book, such as references [1], [14] or [16]
If p is a prime with p ≡ mod, and λ is any fourth-order character mod p, we have τ (λ) + τ
If p is a prime with p ≡ mod, we have the identity p− p−

Summary

Introduction

For any integer m and n, the two-term exponential sum G(k, h, m, n; q) is de ned as q−. Where as usual, e(y) = e πiy, k and h are positive integers with k ≠ h. Many scholars have studied various elementary properties of G(k, h, m, n; q) and obtained a series of results. Weil’s important work [2], one can get the general upper bound estimate p−. Where p is an odd prime, χ is any Dirichlet character mod p and (m, n, p) =. Zhang Han and Zhang Wenpeng [3] proved the identity p− p−.

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Several Lemmas

Proofs of the theorems