Abstract
The main purpose of this paper is using analytic methods and the properties of the Dedekind sums to study one kind hybrid power mean calculating problem involving the Dedekind sums and cubic Gauss sum and give some interesting calculating formulae for it.
1. Introduction
For any integer and integer , the classical cubic Gauss sum is defined as follows:where as usual, and .
This sum plays a very important role in the study of the elementary number theory and analytic number theory, so there are many people who had studied the arithmetical properties of and related contents (see [1–6]). For example, if is an odd prime, Cao and Wang [5] proved the following interesting conclusion: let be an odd prime with . If 3 is not a cubic residue modulo , then we have the identitywhere and is uniquely determined as follows:
On the other hand, we define the Dedekind sums as follows.
Let be a natural number and be an integer prime to . The classical Dedekind sum is defined aswhere
About the various properties of , many authors had studied them and obtained a series of interesting results, and related works can be found in [7–14].
The main purpose of this paper is using the analytic method and the properties of Dedekind sums to study the computational problem of the hybrid power mean:and give some exact computational formulae for (5) with , an odd prime, where and are any two fixed positive integers.
About this problem, so far no one seems to consider; at least, we have not seen any related papers before. Of course, this problem is meaningful, and it can describe the mean value distribution properties of the two different sums. It is clear that if , then from the properties of the complete residue system modulo , we have . This time (5) is meaningless. So, in the following, we only consider the case . The main purpose of this paper is to prove the following several results.
Theorem 1. Let be a prime with . Then, for any positive integers and with , we have the identity
Theorem 2. Let be a prime with . Then, for any positive integer , we have the identitywhere is Legendre’s symbol modulo , denotes any third-order character modulo , denotes the class number of the quadratic field , andsatisfies the third-order linear recursive formula as follows:with the initial values , , and , and , where is uniquely determined by .
Theorem 3. Let be a prime with . Then, for any positive integer , we have the asymptotic formula as follows:where and is a three-order character modulo .
From Theorem 3, we may immediately deduce the following two corollaries.
Corollary 1. Let Let be a prime with , then we have
Corollary 2. Let be a prime with , then we have
2. Several Lemmas
To complete the proofs of our all theorems, we need to prove several simple lemmas. Hereinafter, we shall use some properties of the character sums and Gauss sums, and all of these contents can be found in [15], so they will not be repeated here.
Lemma 1. Let be an odd prime with and be any third-order character modulo , then we have the identitywhere denotes the classical Gauss sums, and , where is uniquely determined by .
Proof. The proof of this lemma is shown in the study by Zhang and Hu [2] or Berndt and Evans [16].
Lemma 2. Let be an odd prime with . Then, for any integer andwe have the third-order linear recursive formula as follows:with the initial values , , and , where is defined as in Lemma 1.
Proof. Let be any third-order character modulo ; then, for any integer , from the properties of the third-order characters and the classical Gauss sums modulo , we have the identityNote that and ; from (16), we haveFrom (16) and Lemma 1, we also haveFrom (17) and the properties of the character sums modulo , we haveCombining (18) and (19), we can deduce thatorSimilarly, from (16), (18), and (21), we haveorIf , then from (18), we haveorNow, Lemma 2 follows from (21)–(25) and .
Lemma 3. Let be a prime with . For any positive integer and any character , if , then we have the identitywhere denotes the principal character modulo .
Proof. Let be a primitive root modulo . Then, from the definition of and the properties of the complete residue system modulo , we haveFrom (27), we haveorIf , then . So, from (29), we have the identityThis proves Lemma 3.
Lemma 4. Let be a prime with and be any third-order character modulo . Then, for any positive integer , we have the identity
Proof. From the properties of the third-order character modulo , we haveFrom (32), the properties of the 3-th residue modulo , and the identity where , we haveThis proves Lemma 4.
Lemma 5. Let be an integer; then, for any integer with , we have the identitywhere denotes the Dirichlet -function corresponding to the character and denotes the summation over all odd characters modulo .
3. Proofs of the Theorems
In this section, we shall complete the proofs of our all theorems. If be a prime, then from Lemma 5, we have
It is clear that if is an integer with , then note that the identities and , and we have
Now, for any positive integer , from (35), we have the identity
If , then for any odd character , if and only if or , where is Legendre’s symbol modulo . In this time, is not an odd character modulo . So, from (37) and Lemma 3, we have
If , then for any odd character , if and only if , or . So, in this time, note that ; applying (37) and Lemma 4, we havewhere is any third-order character modulo .
It is clear that Theorem 1 follows from (36) and (38).
Theorem 2 follows from (39) and Lemma 2.
Now, we prove Theorem 3. Note that and (see [17] or [18]):
From (35), (40), and (41) and Lemma 3, we have the asymptotic formula as follows:
Applying Lemma 2 and mathematical induction, we can easily deduce the estimate
In fact, from Lemma 2, we have , , and . So, (43) is true for , and 2. Note that the third-order linear recursive formula is as follows:
So, for , from the mathematical induction, we can easily deduce the estimate
Combining (42) and (43), we complete the proof of Theorem 3.
4. Conclusion
The main results in this paper are three theorems, which are closely related to Dedekind sums and cubic Gauss sums. They describe that when is a prime, the hybrid power mean of the Dedekind sums and cubic Gauss sums has good mean distribution properties. In fact, we can give some exact calculating formulae (see Theorems 1 and 2) or asymptotic formula (see Theorem 3) for them.
Data Availability
The data generated or analysed during this study are included in this published article.
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. of P. R. China (12161031), the N. S. F. of Inner Mongolia (CN) (2021MS01003), and the Number Theory Research Center of Hetao College.