Abstract
The main purpose of this paper is using the analytic method, A. Weil’s classical work for the upper bound estimate of the general exponential sums, and the properties of Gauss sums to study the hybrid mean value problem involving Dedekind sums and the general exponential sums and give a sharp asymptotic formula for it.
1. Introduction
For a positive integer and an arbitrary integer , the classical Dedekind sums are defined by where The various properties of were investigated by many authors; see [1–5]. For example, Conrey et al. [3] studied the mean value distribution of and proved the asymptotic formula where denotes the summation over all such that , and
Zhang [6] established a close contact between and the mean square value of Dirichlet -functions (see Lemma 4). Maybe the most important property of is its reciprocity theorem (see [2]). That is, for all positive integers and with , we have the identity
On the other hand, Liu and Zhang [5] studied the hybrid mean value problem involving and and proved the following conclusion.
Let be a square-full number; then where is the Ramanujan sum, defined as (see Theorem 8.6 of [7]) and is the famous Möbius function.
Weil [8] studied the upper bound estimate of the general exponential sums and proved the estimate (see Corollary 2F of [9, page 45]) where , is an odd prime, denotes the big- constant which depends only on , and is th integral coefficients polynomial with .
In fact, the estimate (9) is the best one, since if , then from Gauss famous work (see [7, page 195]) we have
The content and form of this paper are different from the references [3, 6]. Conrey et al. [3] studied the general th power mean of Dedekind sums and obtained an asymptotic formula. Zhang [6] only obtained a relationship between and the mean square value of Dirichlet -functions. Our work is using Zhang’s result, Weil’s classical work for the upper bound estimate of the general exponential sums, and the properties of Gauss sums to study the hybrid mean value problem involving Dedekind sums and the general exponential sums and give a sharper asymptotic formula for it.
2. Main Theorems
In this paper, we will obtain the following two results.
Theorem 1. Let be an odd prime and let be any fixed positive integer. Suppose that is a polynomial with integral coefficients having and . Then we have the asymptotic formula where denotes the solution of the congruence equation .
For the classical Kloosterman sums, we can also obtain a similar conclusion. That is, we have the following.
Theorem 2. Let be an odd prime; then we have the asymptotic formula where denotes the classical Kloosterman sums.
For general integer , whether there exists a similar asymptotic formula as in Theorem 1 (or Theorem 2) is an open problem.
3. Lemmas and Proofs of the Theorems
In order to complete the proof of our theorems, we need the following several simple lemmas. Hereinafter, we will use many definitions and properties of Gauss sums, Kloosterman sums, and character sums, all of which can be found in [7, 10–13], so they will not be repeated here. First we have the following.
Lemma 3. Let be an odd prime and let be the Dirichlet character . Then one has the estimate where denotes the summation over all odd characters .
Proof. From the method of proving Lemma 5 in [14] we may immediately deduce this estimate.
Lemma 4. Let be an integer; then for any integer with , one has the identity where denotes the Dirichlet -function corresponding to character .
Proof. See Lemma 2 of [6].
Lemma 5. Let be an odd prime; then one has the asymptotic formula
Proof. See the theorem and corollary of [14].
Proof of Theorem 1. From Lemma 4 with , an odd prime, we have From the definition of and the properties of Gauss sum, we have Note that for any nonprincipal character , we have . For any integer , since , the polynomial satisfying . Applying (9), we have the estimate Now combining (16), (17), (18), (19), and Lemma 3, we have the asymptotic formula This proves Theorem 1.
Proof of Theorem 2. Note that if , then . Consider From (16), the properties of Gauss sums, and Lemma 5, we have This completes the proof of Theorem 2.
Conflict of Interests
The authors declare that they have no conflict of interests.
Acknowledgments
The authors would like to thank the referees for their very helpful and detailed comments, which have significantly improved the presentation of this paper. This work is supported by the Specific Fundation of Xi’an University of Technology (no. 2013TS011) and the Specific Fundation for Talents of Northwest A&F University.