The Scientific World Journal

Volume 2014 (2014), Article ID 726053, 5 pages

http://dx.doi.org/10.1155/2014/726053

Research Article

On the Generalization of Lehmer Problem and High-Dimension Kloosterman Sums

Guohui Chen¹ and Han Zhang²

¹College of Mathematics and Statistics, Hainan Normal University, Hainan 571158, China

²School of Mathematics, Northwest University, Xi’an, Shaanxi 710127, China

Received 1 April 2014; Accepted 8 July 2014; Published 16 July 2014

Academic Editor: Kinkar Ch Das

Copyright © 2014 Guohui Chen and Han Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

For any fixed integer and integer with , it is clear that there exist k integers such that . Let denote the number of all such that and 2†. In this paper, we will use the analytic method and the estimate for high-dimension Kloosterman sums to study the asymptotic properties of and give two interesting asymptotic formulae for it.

1. Introduction

Let be an odd number. For each integer with , it is clear that there exists one and only one with such that . Let denote the number of all in which and are of opposite parity. Professor D. H. Lehmer [1] asked us to study or at least to say something nontrivial about it. It is known that when . Some works related to the Lehmer problem can be found in references [2–5]. For example, Zhang [2, 4] proved the asymptotic formula

In this paper, we will study a new summation related to the Lehmer problem. For any fixed integer and integer with , we define the sums as follows:

In fact, is a generalization of the Lehmer problem. For example, if and , then from the definition of we have

So becomes , the Lehmer problem.

Now we are concerned about the arithmetical properties of . This problem is interesting, because it is a generalization of the Lehmer problem.

In this paper, we use the analytic method and the estimate for high-dimension Kloosterman sums to study the asymptotic properties of and give two interesting asymptotic formulae for it. That is, we will prove the following.

Theorem 1. Let be an odd prime. Then for any fixed integer and integer with , we have the asymptotic formula

In order to facilitate the description of Theorem 2, we need to give the definition of high-dimension Kloosterman sums . Let be an integer. For any integer , we define

where .

About some arithmetical properties of , one can find them in [6–8]. Let denote the error term in the asymptotic formula of . As another main content of this paper, we will study the asymptotic properties of the hybrid mean value of and and also give a sharp asymptotic formula for it. That is, we will prove the following.

Theorem 2. Let be an odd prime. Then for any fixed integer , we have the asymptotic formula

where , denotes any fixed positive number.

The constants in Theorem 2 cannot be omitted. Otherwise, the main term in Theorem 2 is zero. If and , then from Theorem 2 we can also deduce the following two corollaries.

Corollary 3. Let be an odd prime. Then for any fixed positive number , we have the asymptotic formula

Corollary 4. Let be an odd prime. Then for any fixed positive number , we have the asymptotic formula

2. Several Lemmas

In this section, we will give several lemmas, which are necessary in the proofs of our theorems. Hereinafter, we will use many properties of Gauss sums and the estimate for high-dimension Kloosterman sums; all of these contents can be found in references [6, 9], so they will not be repeated here. First we have the following.

Lemma 5. Let be an odd prime. Then for fixed integer and any integer , we have the estimate

Proof. See [6, 7].

Lemma 6. Let be an odd prime. Then for any odd character (i.e., ), we have the identity

Proof. See [10] or Lemma  3 in [3].

Lemma 7. Let be an odd prime. Then for any integer , we have

where denotes the summation over all odd characters , denotes the classical Gauss sums, and denotes the Dirichlet -function corresponding to .

Proof. From the orthogonality of characters and the definition of we have the identity

For any odd character , from Theorems 12.11 and 12.20 of [9] we have

Note that, for any even character , we have the identity

from (13) and Lemma 6 we have

Now Lemma 7 follows from (12) and (15).

Lemma 8. Let be an odd prime and a fixed integer with . Then for any nonprincipal character and any real numbers , we have the estimate

Proof. We use mathematical induction to prove this lemma. If , then from the Pòlya-Vinogradov inequality we have

Assume that the lemma holds for . That is,

Then for , note that ; applying estimate (18) and the Pòlya-Vinogradov inequality we have

Now our lemma follows from the induction.

3. Proofs of the Theorems

In this section, we will prove our conclusions. First we prove Theorem 1. For any real number , applying Abel’s identity (see Theorem 4.2 of [9]) we have

For any integer , from Lemma 5 and the definition of we have

Applying (20) and the binomial expression we have the estimate

Taking , note that and the identity

and applying Lemma 8 we have the estimate

Combining (20), (22), (24), and Lemma 7 we may immediately deduce the asymptotic formula

The proof of Theorem 1 is right.

Now we prove Theorem 2. For any nonprincipal character , from the definition and properties of Gauss sums we have

Note that ; from Lemma 7 and the definition of we have

where , , denotes any fixed positive number and denotes the th divisor function. That is, .

The proof of Theorem 2 is right.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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 P. S. F. (2013JZ001) and N. S. F. (11371291) of China.

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