Research Article | Open Access
On the Generalization of Lehmer Problem and High-Dimension Kloosterman Sums
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.
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  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
Lemma 6. Let be an odd prime. Then for any odd character (i.e., ), we have the identity
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  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 ) we have For any integer , from Lemma 5 and the definition of we haveApplying (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 haveNote 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.
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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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.