#### Abstract

The 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.

#### 1. Introduction

Let be a fixed integer. For any integers and with , the two-term exponential sums and quartic Gauss sums are defined bywhere, as usual, and .

These sums play a significant role in the research of analytic number theory, and many number theory problems are closely related to them. Therefore, for the sake of promoting the development of research work in related fields, it is necessary to study the various properties of and . Some research results in these fields can be found in references [1–12]. We will not list all of them. For example, Zhang and Zhang [1] proved the identitywhere be an odd prime and denotes any integer with .

Zhang and Han [2] obtained the identity:where denotes an odd prime with .

Zhang and Zhang [3] derived that, for any prime , one has the identity:where, as usual, denotes the Legendre’s symbol modulo , , and is an integer which satisfies the estimate .

The author [4] studied the following hybrid power mean:and obtained two interesting fourth-order linear recurrence formula as follows:(1), where the first four item in sequence is ; ; ; and (2)

The first four items in the sequence is ; ; ; and .

Inspired by references [1–4], in this paper, we will consider the following -th hybrid power mean:and -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 and a fourth-order linear recurrence formula for by using the properties of Legendre’s symbol and the classical Gauss sums. That is, we will prove the following results.

Theorem 1. *If is an odd prime with , for any four-order character , we havewhere denotes the classical Gauss sums.*

Theorem 2. *If is an odd prime with , then we have the second-order linear recurrence formula:with the initial valueswhere denotes Legendre’s symbol modulo . and are two integers, and they satisfy the estimates and , .*

Theorem 3. *If is an odd prime with , then we have the fourth-order linear recurrence formula:with the initial values*

From (4), Theorem 1, and Weil’s works [13,14], we have the estimates:

Applying (2), Theorems 2 and 3, the properties of the linear recursive sequences, and these three estimates, we can deduce the following three corollaries.

Corollary 1. *If is an odd prime with , for any positive integer , we have the asymptotic formula:where denotes the big - constant depending only on the positive integer .*

Especially for , we have the asymptotic formula:

Corollary 2. *If is an odd prime with , for any positive integer , we have the asymptotic formula:*

Corollary 3. *If is an odd prime with , then we have the asymptotic formula:*

#### 2. 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. They can be found in references [15–17]. Firstly, we have the following:

Lemma 1. *If is an odd prime with , for any four-order character , we have*

*Proof. *Let denotes the Legendre’s symbol modulo . Then, for any integer , from the properties of the Legendre’s symbol modulo , we haveNote that and , and using the definition and properties of Gauss sums, reduced residue system modulo , and formula (19), we haveNote that , and from the properties of Gauss sums, we haveFrom (20) and (21), we haveThis proves Lemma 1.

Lemma 2. *If is an odd prime with , for any four-order character , we have*

*Proof. *From Lemma 1, we haveLet , then from the properties of the complete residue system modulo , we haveFrom (19) and the methods of proving (20), we haveFrom (19), we haveCombining (24)–(27), we have the identityThis proves Lemma 2.

Lemma 3. *If is an odd prime with , then we have*

*Proof. *From the properties of the classical Gauss sums and the methods of proving Lemmas 1 and 2, we haveLet and , then from the properties of the complete residue system modulo , we haveFrom (19), we haveFrom (19), we also haveNow, combining (27)–(33), we have the identity:This proves Lemma 3.

Lemma 4. *Let is an odd prime with , then for any four-order character , we have the identity:where is an integer.*

*Proof. *See Lemma 2.2 in Han [5].

#### 3. Proofs of the Theorems

In this section, we will complete the proofs of our theorems. In fact, for any prime with , from Lemmas 2 and 3, we have

This proves Theorem 1.

Now, we prove Theorem 2. If , then for any four-order character , we have . From the properties of the Gauss sums and the four-order character modulo , we have

So, from (37), (38), and Lemma 4, we have

So, for any integer , from (2), (4), (39), and (40), we havewith the initial valueswhere is the same as the definition in (4). This proves Theorem 2.

Now, we prove Theorem 3. If , then note that , so from (37) and Lemma 4, we have

So, for any integer , from (43) and (44), we havewith the initial values

This proves Theorem 3.

Now, we will give a simple proof for Corollary 2. Note the estimateand the identity

Note that the linear recurrence sequence has the general terms:

If and , then we have and

Combining (50) and (52), we have the asymptotic formula:

This completes the proofs of our all results.

#### 4. Conclusion

The main results of this paper are two linear recurrence formulas for the hybrid power means of the two-term exponential sums and the quartic Gauss sums . As some applications of these results, we obtained a sharp asymptotic formula for the hybrid power means:for all positive integer and odd prime with . All these results are new contributions to the related fields.

#### Data Availability

Our field is theoretical mathematics, and we do not use any data.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

#### Authors’ Contributions

All authors have equally contributed to this work. All authors read and approved the final manuscript.

#### Acknowledgments

This work was supported by the N.S.F. (11771351) of China, the Scientific Research Program funded by Shaanxi Provincial Education Department (program no. 19JK0978), the Natural Science Basic Research Plan in Shaanxi Province of China (2019JM-207), and the Doctoral Scientific Research Project of Xi’an Aeronautical University.