Abstract

In this paper, we use the mean value theorem of Dirichlet -functions and the properties of Gauss sums and Dedekind sums to study the hybrid mean value problem involving Dedekind sums and the general Kloosterman sums and give an interesting identity for it.

1. Introduction

Let be a natural number and be an integer coprime to . The classical Dedekind sumswheredescribes the behaviour of the logarithm of the -function (see [1, 2]) under modular transformations. There are many papers written on their various properties (see the examples in [3–10] and [11]).

In particular, Zhang and Liu [12] studied the hybrid mean value problems related to Dedekind sums and Kloosterman sums:where is an integer, denotes the summation over all with , , and denotes the multiplicative inverse of . They proved the following results:

Theorem 1. Let be an odd prime, then one has the identitywhere denotes the class number of the quadratic field .

Theorem 2. Let be an odd prime, then one has the asymptotic formula:where .

It is natural that people will ask, for the general Kloosterman sumswhat will happen? Whether there exists an identity similar to Theorem1? Here, denotes any Dirichlet character .

The main purpose of this paper is to answer these questions. That is, we shall use the mean value theorem of Dirichlet -functions and the properties of Gauss sums and Dedekind sums to prove the following.

Theorem 3. Let be an odd prime with . Then, for any Dirichlet character , we have the identity:

Theorem 4. Let be an odd prime with . Then, for any Dirichlet character , we have the identity:where denotes the Legendre symbol and denotes the class number of the quadratic field .

It is clear that if , then . Note that , from Theorems 3 and 4, we may immediately deduce Theorem1 in [12], so our results are the generalization of [12].

2. Several Lemmas

In this section, we shall give several simple lemmas, which are necessary to the proofs of our theorems. Hereafter, we shall use many properties of character sums and Gauss sums, and all of these can be found in reference [13]. First, we have the following.

Lemma 1. Let be a prime, be any fixed Dirichlet character . Then, for any nonprincipal character with , we have the identity:where denotes the principal character , denotes the Gauss sums defined as , and denotes the complex conjugate of .

Proof. From the definition of Kloosterman sums and the properties of Gauss sums, we haveOn the other hand, from the properties of Gauss sums, we haveCombining (10) and (11), we may immediately deduce the identity:This proves Lemma 1.

Lemma 2. Let be a prime with and be any odd character . Then, we have the identity:

Proof. Since and is an odd character , we know is not the Legendre symbol and . Note that , from (10), we haveThis proves Lemma 2.

Lemma 3. Let be a prime with and be the Legendre symbol. Then, we have the identity:

Proof. Note that , , and , and from the definition of and the properties of Gauss sums, we haveThis proves Lemma 3.

Lemma 4. Let be an integer, then for any integer with , we have the identity:where denotes the Dirichlet -function corresponding to the character .

Proof. See Lemma 2 of [7].

3. Proof of the Theorems

In this section, we will complete the proof of our theorems. First we prove Theorem 3. From Lemma 4 and the definition of , we haveand (with )

Since , we know the Legendre symbol is an even character . Note that, for any nonprincipal character , . So, if is an even character , then from Lemma 1, (18), and (19), we have

If is an odd character , then note that the identity:from (18), (19), Lemmas 1 and 2, and the method of proving (20), we havewhere denotes the summation over all odd characters with .

Combining (20) and (22), we may immediately deduce the identity:

This proves Theorem 3.

Now, we prove Theorem 4. Since , we know the Legendre symbol is an odd character and . If is an even character , then note that (see reference [14]), from (18), (19), Lemma 1 and the properties of Gauss sums, we havewhere denotes the class number of the quadratic field .

If is an odd nonreal character , then from (18), (19), Lemmas 1 and 2, and the method of proving (22) and (24), we have

If is the Legendre symbol, then from (18), (19), Lemmas 1 and 3, and the method of proving (25), we have

Combining (24), (25), and (26), we can deduce the identity:

This completes the proof of Theorem 4.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by X. M. U. F. (nos. 2016PT38, 2017GJFY21, and 2018XNRC05) and P. N. S. F. (no. 2019JM573) of PR China.