Some New Identities Related to Dedekind Sums Modulo a Prime

Abstract

The main purpose of this article is to use some identities of the classical Gauss sums, the properties of character sums, and Dedekind sums (modulo an odd prime) to study the computational problem of one-kind mean values related to Dedekind sums and give some interesting identities for them.

1. Introduction

To describe the results of this paper, we first need to introduce the Drichlet character and some famous sums in the analytic number theory. Let be an integer and be the group of reduced residue classes modulo . Corresponding to each character of , we define an arithmetical function modulo as follows:

In addition to periodicity, the Drichlet characters modulo also have orthogonality which can be found in reference [1]. For any Dirichlet character modulo , the classical Gauss sums is defined aswhere is any integer, , and .

For convenience, we write . About some properties of , many scholars have studied them and obtained a series of important results. Perhaps, the most important properties of are the following two conclusions:(i)If , then for any character modulo , we have the identity (see [1, 2])(ii)If is any primitive character modulo , then for any integer , one has also and the identity .

In addition, for any prime with and any fourth-order primitive character modulo , Chen and Zhang [3] studied the properties of and proved the identitywhere , and denotes the Legendre’s symbol modulo .

Here, the constant in (4) has a special meaning. In fact, if prime , then we have the identity (for this, see Theorems 4–11 in [4]).where is any quadratic nonresidue modulo . That is, .

Chen [5] used the analytic methods to obtain another identity for the six-order primitive characters modulo . That is, she proved the following conclusion. Let be a prime with , then one has the identitywhere , is uniquely determined by , and .

There are many other related results, and we will not list them all here.

Obviously, the identities (5) and (6) look very concise and beautiful, but whether they can be applied in theory or practice is what we care most. Recently, we have found that these identities can be used to calculating some mean value problems of the Dedekind sums. And for that, we need to introduce the definition of the Dedekind sums. For any integers and , the classical Dedekind sums is defined as follows (see [6]):where as usual,

In fact, this sums describes the behaviour of the logarithm of the eta-function (see [7, 8]) under modular transformations. Because of the importance of in the analytic number theory, many authors have studied the arithmetical properties of and obtained many interesting results, some of them can be found in [9–15]. Some relevant and meaningful work can also be found in [16–18]. In order to avoid the tedious, we do not want to list them one by one. Maybe the most important properties of are its reciprocity theorem (see [6, 9]). That is, for all positive integers and with , we have the identity

Rademacher and Grosswald [8] also obtained a three-term formula similar to (9).

The main purpose of this paper is to study the calculating problems of one kind mean values of . That is,where and are two positive integers.

This work is mainly because the high dimensional sums such as the -dimensional Kloosterman sums and -dimensional character sums play an important role in the research of number theory. For example, Li and Zhang [19] study the sumswhere is an odd prime, is any nonprincipal Dirichlet character , is any fixed positive integer, and is any integer. Also, they obtained the following conclusions.

Theorem 1. Let be an odd prime, is an integer with . Then, for any nonprincipal character , one has the identity

Theorem 2. Let be an odd prime, is an integer with , and is any nonprincipal character . If is a -th character (that is, there exists a character such that ), then one has

Hence, it is meaningful in further exploring the problem of value distribution of on certain special sets. It may be possible to characterize some profound properties of .

In this paper, we give some accurate calculating formulas for with and or and . That is, we use the identities (5) and (6) of the classical Gauss sums and analytic methods to prove the following three interesting conclusions.

Theorem 3. Let be an odd prime, then we have the identitieswhere denotes the class number of the imaginary quadratic field , denotes any four-order primitive character modulo , and are defined as in (5), and denotes the Dirichlet -function corresponding to character modulo .

Theorem 4. Let be an odd prime, then we have the identitieswhere denotes any six-order primitive character modulo , , and is uniquely determined by .

Theorem 5. Let be an odd prime with , then we have the identities

Some notes: since , so for any odd number , we have identities

Therefore, we only consider the case with an even number of variables.

If in Theorem 5, then the situation is more complicated, and we cannot yet get accurate calculation results.

In addition, whether these sums have reciprocal laws is also an interesting problem.

These will be the subjects of our further research.

2. Several Lemmas

In this section, we will deduce several simple lemmas that are necessary in the proofs of our main results. Hereinafter, we shall use the knowledge of the analytic number theory, and the properties of the classical Gauss sums and Dedekind sums, all these can be found in references [1, 2, 4, 6]. Therefore, we do not repeat them here. First, we have the following:

Lemma 6. Let be an odd prime. Then, for any odd character modulo , we havewhere denotes the four-order primitive character modulo .

Proof. From the definition of the classical Gauss sums, the properties of the trigonometric sums, and note that , and the identitywe haveIf and , then we haveIf , let denote any four-order primitive character modulo . That is, , the principal character modulo and for . Note that , , if and , if . So, in these cases, from the properties of the characters modulo , we haveNote that , from (11), (20), and (21), we have the identitiesThis proves Lemma 6.

Lemma 7. Let be an odd prime. Then, for any odd character modulo , we have the identities

Proof. From the methods of proving (11), we haveIf , then for any , we haveIf , let denote any six-order primitive character modulo , where denotes any three-order primitive character modulo . Then note that , from the properties of the characters modulo we haveIf or , then for any character modulo , we haveNote that , from (22), (25)–(27), we haveThis proves Lemma 7.

Lemma 8. Let be an odd prime with . Then, for any odd character modulo , we havewhere denotes the four-order primitive character modulo and .

Proof. From the definition of the classical Gauss sums, the properties of the trigonometric sums, and note that , and the identitywe haveIf , let , then we haveNow, Lemma 8 follows from (28) and (32).

Lemma 9. Let be an integer, then for any integer with , we have the identitywhere denotes the Dirichlet -function corresponding to .

Proof. See Lemma 2 in [10].

3. Proofs of the Theorems

In this section, we will provide the proofs of our theorems. We start with the proof of Theorem 3. For any odd prime and integer , from Lemma 9, we have

If , note that and , and from (33) and Lemma 6, we have

If , note that and there are two four-order primitive characters and modulo such that and . So in this case, from (4), (5), and (33) and Lemma 6, we have

If , then for any odd character , we have . So in this case, from Lemma 6 and the methods of proving (35), we have the identity

Now, Theorem 3 follows from (35)–(37).

Now, we prove Theorem 4. If , then for any , we have

Note that , and from (33) and Lemma 7, we have

If , let denote any six-order primitive character modulo , where denotes any three-order primitive character modulo . Then, note that there are two six-order primitive characters , modulo , , and , and from (6) and (33), Lemma 7, and the methods of proving (39), we have

If or , then for any odd character modulo , we have

This time, we have the identity

Now, Theorem 4 follows from (39)–(41).

Now, we prove Theorem 5. If , note that and there are two four-order primitive characters and modulo such that , , and for any integer with . So in this case, from (33), Lemma 8, and the methods of proving Theorem 3, we may immediately deduce Theorem 5.

This completes the proofs of our all results.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

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

Acknowledgments

The author would like to thank the referees for their very helpful and detailed comments. This work was supported by the N. S. F. (12161031) of P. R. China, the N. S. F. (2021MS01003) of Inner Mongolia (CN), and the Higher Education Scientific Research Project (NJZY21173) of Inner Mongolia (CN).