One-Kind Hybrid Power Means of the Two-Term Exponential Sums and Quartic Gauss Sums

Xiaoxue Li and Li Chen 2 School of Science, Xi’an Aeronautical University, Xi’an, Shaanxi, China School of Mathematics, Northwest University, Xi’an, Shaanxi, China Correspondence should be addressed to Li Chen; chenli_0928@163.com Received 13 December 2020; Accepted 2 April 2021; Published 27 April 2021 Academic Editor: Tingting Wang Copyright © 2021 Xiaoxue Li and Li Chen. +is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. +e main purpose of this article is using the analytic methods and the properties of the classical Gauss sums to study the calculating problem of the hybrid power mean of the two-term exponential sums and quartic Gauss sums and then prove two interesting linear recurrence formulas. As applications, some asymptotic formulas are obtained.


Introduction
Let q ≥ 3 be a fixed integer. For any integers k ≥ 2 and m with (m, q) � 1, the two-term exponential sums G(m, k; q) and quartic Gauss sums G(m, q) are defined by G(m, k; q) � q−1 a�0 e ma k + a q , where, as usual, e(y) � e 2πiy and i 2 � −1. ese sums play a significant role in the research of analytic number theory, and many number theory problems are closely related to them. erefore, for the sake of promoting the development of research work in related fields, it is necessary to study the various properties of G(m, k; q) and G(m, q). Some research results in these fields can be found in references [1][2][3][4][5][6][7][8][9][10][11][12]. We will not list all of them. For example, Zhang and Zhang [1] proved the identity where p be an odd prime and n denotes any integer with (n, p) � 1. Zhang and Han [2] obtained the identity: where p denotes an odd prime with 3∤(p − 1). Zhang and Zhang [3] derived that, for any prime p, one has the identity: where, as usual, ( * /p) � χ 2 denotes the Legendre's symbol modulo p, d · d ≡ 1modp, and δ � e author [4] studied the following hybrid power mean: and obtained two interesting fourth-order linear recurrence formula as follows: Inspired by references [1][2][3][4], in this paper, we will consider the following 2k-th hybrid power mean: and k-th hybrid power mean Of course, the work in this paper looks a little imaginative with that in [4], but they have different essence, and the main difference lies in the power of the two-term exponential sums. In fact, it is a lot easier that we are dealing with the quadratic power mean of the two-term exponential sum in [4]. In this paper, we are dealing with the fourth power of the two-term exponential sums, and it is very difficult.
In this paper, we will give a second-order linear recurrence formula for A k (p) and a fourth-order linear recurrence formula for B k (p) by using the properties of Legendre's symbol and the classical Gauss sums. at is, we will prove the following results. Theorem 1. If p > 3 is an odd prime with p ≡ 1mod4, for any four-order character χ 4 modp, we have where τ(χ) � p−1 a�1 χ(a)e(a/p) denotes the classical Gauss sums.
Theorem 2. If p > 3 is an odd prime with p ≡ 5mod8, then we have the second-order linear recurrence formula: with the initial values

two integers, and they satisfy the estimates |α|
Theorem 3. If p > 3 is an odd prime with p ≡ 1mod8, then we have the fourth-order linear recurrence formula: with the initial values From (4), eorem 1, and Weil's works [13,14], we have the estimates: Applying (2), eorems 2 and 3, the properties of the linear recursive sequences, and these three estimates, we can deduce the following three corollaries.

Corollary 1.
If p is an odd prime with p ≡ 5mod8, for any positive integer k, we have the asymptotic formula: where O k denotes the big -O constant depending only on the positive integer k.
Especially for k � 2, we have the asymptotic formula:

Several Lemmas
In this section, we will give four basic lemmas that they are all necessary in the proofs of the theorems. Certainly, the proofs of these lemmas need some theoretical knowledge of elementary and analytic number theory. ey can be found in references [15][16][17]. Firstly, we have the following: is an odd prime with p ≡ 1mod4, for any four-order character χ 4 modp, we have Proof. Let χ 2 denotes the Legendre's symbol modulo p. en, for any integer b, from the properties of the Legendre's symbol modulo p, we have Note that χ 3 4 � χ 4 and τ(χ 2 ) � � � p √ , and using the definition and properties of Gauss sums, reduced residue system modulo p, and formula (19), we have 4
□ Lemma 2. If p > 3 is an odd prime with p ≡ 1mod4, for any four-order character χ 4 modp, we have Proof. From Lemma 1, we have Let c � b − 1, then from the properties of the complete residue system modulo p, we have Combining (24)-(27), we have the identity is proves Lemma 2.