On the Fourth Hybrid Power Mean Involving the Generalized Gauss Sums

Abstract

In this paper, we use the elementary and analytic methods to study the fourth hybrid power mean involving the generalized Gauss sums and prove several interesting identities for them.

1. Introduction

Let be an integer, the generalized th Gauss sums is defined as follows:where is a Dirichlet character modulo and and are any integers with , , and .

is used to solve many problems related to number theory. Thus, many scholars at home and abroad have conducted various research studies on . When , is called as classical Gauss sums and represented by . The most classical and important property of is that if is a primitive character modulo , then and (see [1,2]). Besides, some mathematicians have obtained identities related to classical Gauss sums. For example, when is a prime number, . Berndt and Evans [3] obtained the following identity, i.e.,where is any third-order character modulo , is uniquely determined by , and .

When is a prime number, . Chen and Zhang [4] obtainedwhere is any fourth-order character modulo , , and denotes Legendre’s symbol modulo .

Furthermore, and go hand in hand. When , can be written as the sum of two square numbers (see Theorem 4–11 in [5]), that is,where is any quadratic nonresidue modulo , i.e., .

There are a number of studies with regard to the Gauss sums and their recursion properties, which can be found in [6–15].

Understanding the relationship between different th Gauss sums will help us have a deeper understanding of Gauss sums. Thus, the aim of this paper is to consider the hybrid power mean of and , i.e.,where is a prime number with , and denote any fourth-order and third-order character modulo , respectively, and and are two nonnegative integers.

So far, we have not seen anyone studying the above problem. Thus, in this paper, using analytic methods and the properties of the Gauss sums, we obtain several interesting computational formulae for (5) with and and . That is, we give the proof of the following two results.

Theorem 1. Let be an odd prime number with . Then, for any third-order character and fourth-order character modulo , we have the identity as

Theorem 2. Let be an odd prime number with . Then, for any fourth-order character modulo , we have the identity aswhere the definition of is the same as in (2), or , and the definitions of and are the same as defined in (4).

Obviously, Theorem 2 has its limitations. That is, it is not clear what value is. How to determine the exact value of in Theorem 2 is an interesting open problem. We will start to study above works in the future.

2. Several Lemmas

In this section, we are going to give five lemmas, which are very important in the proof of the theorems. In the process of proof, we will use some knowledge of analytic number theory, which can be found in [1, 2, 5].

Lemma 1. Let be a prime number with . Then, for any sixth-order character , we have the identity aswhere , is uniquely determined by , and .

Proof. For this, see Chen [16].

Lemma 2. Let be an odd prime number with . Then, for any character modulo , we have the identity aswhere is a third-order character modulo .

Proof. For this, see [17] or [18]. The general result can also be found in [19].

Lemma 3. Let be a prime number with , be any fourth-order character, and be any third-order character modulo . Then, we have the identity as

Proof. Note that and . From the properties of the classical Gauss sums, we obtainOn the other hand, we havewhere . Note that and , and we also haveFrom (11) and (13) and noting that , we haveorSince and ,Combining (15), (16), and Lemma 1, we have the identity asThis proves Lemma 3.

Lemma 4. Let be a prime number with , be any fourth-order character, and be any third-order character modulo . Then, we have the identitywhere the definitions of and are given in (4).

Proof. Note that , and taking in Lemma 2, we haveSo, from Lemma 3 and (19), we haveFrom Lemma 3, we haveNote that . Combining (20), (21), and (3), we have the identity asFrom (22), we can obtain Lemma 4.

Lemma 5. Let be an odd prime number with . Then, for any third-order character and fourth-order character modulo and any integer with , we have the identity as

Proof. Using the properties of the Gauss sums and Dirichlet character, we haveNote that , , and . From (24) and (25), we haveThis proves Lemma 5.

3. Proofs of the Theorems

Now, let us begin to prove the theorems. Let be an odd prime number with , and then, for any nonprincipal character modulo , we have

From (27) and Lemma 5, we have

This proves Theorem 1.

Now, we prove Theorem 2. Note that and , and from (24), we have

Applying Lemma 2, we have

From (27), (29), (30), and Lemma 4, we have

This completes the proof of Theorem 2.

4. Conclusion

The main results of this article are two theorems. Theorem 1 studied an exact identity for the hybrid power mean involving two generalized Gauss sums. Theorem 2 studied one kind of fourth power mean of the generalized Gauss sums and proved that it has four possibilities. These results, on the one hand, give the exact values of one kind of hybrid power mean of the generalized Gauss sums, and on the other hand, they give new contribution to the research of related fields.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

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

Authors’ Contributions

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

Acknowledgments

This work was supported by the N. S. F. of P. R. China (11771351 and 12126357).