Abstract
The main purpose of this paper is to use the elementary and analytic methods, the properties of Gauss sums, and character sums to study the computational problem of a certain hybrid power mean involving the Dedekind sums and a character sum analogous to Kloosterman sum and give two interesting identities for them.
1. Introduction
We all know that the classical Dedekind sums is defined (see [1]) aswhere is a positive integer, is any integer prime to , and
This sum describes the behaviour of the logarithm of the eta function under modular transformations, see [1, 2], for related references. Because of the importance of this sum in analytic number theory, many scholars have studied its various properties and obtained a series of important results. Perhaps, the most important property of is its reciprocity theorem (see [3]). That is, for any positive integers and with , one has the identity
Some other papers related to Dedekind sums can be found in [4–6], and we do not want to list them all here.
On the contrary, we also introduce another character sums analogous to Kloosterman sums as follows. For any integer , let be a Dirichlet character . For any positive integer and integer , we definewhere denotes the summation over all such that and denotes the inverse of . That is, .
About the properties of , some people had studied it and obtained some important results. For example, from the very special case of Weil’s work [7] one can obtain the estimatewhere is a prime and is any integer. Some related important works can also be found in [7–11].
In this paper, we consider the computational problem of the hybrid power mean involving the Dedekind sums and . That is,
However, for this hybrid power mean, it seems that none has studied it yet; at least, we have not seen any related results before. The problem is interesting because it is closely related to Dirichlet L-functions. In fact, for some special positive integers , we can give an exact computational formula for (6). The main work of this paper is to reveal this point. That is, we shall use the elementary and analytic methods, and the properties of character sums to prove the following two conclusions.
Theorem 1. Let be an odd prime with , and denotes Legendre’s symbol . Then, for any positive integer with , we have the identitywhere denotes the class number of the quadratic field .
Theorem 2. Let be an odd prime with and be any positive integer with . Then, we have the identityIf be an odd prime with , then we have the identitywhere denotes the fourth-order character , is an integer, and denotes the Dirichlet -function corresponding to .
Taking in Theorem 1 and Theorem 2, then we have the following.
Corollary 1. Let be an odd prime with and ; then, we have the identity
Corollary 2. Let be an odd prime with ; then, we have
Notes: Obviously, in a sense, Corollary 1 gives us efficient methods to compute the class number that can be done on a computer.
It is easy to prove that if , then, for any positive integer , we have
If , then, for any positive integer , we also have
For general composite number , whether there exists an exact computational formula for (6) will be our further research problem.
2. Several Lemmas
In this section, we shall give several simple lemmas, and they are necessary in the proofs of our theorems. First, we have the following.
Lemma 1. Let be a prime, and and are two nonprincipal characters with . Then, for any positive integer , we have the identitywhere denotes the classical Gauss sums.
Proof. For any integer and nonprincipal character , from the properties of Gauss sums (see Theorem 8.20 in [12]), we haveUsing (15) and the properties of the reduced residue system , we haveSo, with the repeated use of (15) in (16), we haveThis proves Lemma 1.
Lemma 2. Let be an integer; then, for any integer with , we have the identitywhere denotes the Dirichlet -function corresponding to character .
Lemma 3. If is a prime with and is any fourth-order character , then we have the identitywhere is an integer.
Proof. See Lemma 2.2 of [13] or Lemma 3 of [14].
Lemma 4. If is a prime with and is any fourth-order character , then, for any positive integer , we have the identity
Proof. First, for all nonnegative integers and real numbers and , we have the identitywhere denotes the greatest integer . This formula is obtained because of Waring [15]. It can also be found in [16].
Note that if , then, for any fourth-order character , we have . So, . Thus, taking and , from (4) and Lemma 3, we haveThis proves Lemma 4.
3. Proofs of the Theorems
In this section, we shall complete the proofs of our theorems. First, we prove Theorem 1. From Lemma 2, we have
If is a prime with , then, for any positive integer with , let ; then, must be an odd number. If characters and satisfy with , then . If is Legendre’s symbol , then we have . Note that if , then , and . So, from (23) and Lemma 1, we have
This proves Theorem 1.
Now, we prove Theorem 2. Let be a prime with ; then, . For any four-order character , we have . So, for any positive integer with , note that is an odd number; if , then . In this time, must be a fourth-order character and ; from (23), Lemma 1, and Lemma 4, we have
If , then , so in this time, we have
Combining (25) and (26), we may immediately deduce Theorem 2.
Data Availability
No data were used to support this study.
Conflicts of Interest
The author declares that there are no conflicts of interest regarding the publication of this paper.
Authors’ Contributions
Xu Xiaoling has contributed to this work and read and approved the final manuscript.
Acknowledgments
This work was supported by the N. S. F. of China (11771351) and the 2019 Special Scientific Research Program of Shaanxi Provincial Department of Education (19JK0978).