High-Dimensional D. H. Lehmer Problem over Quarter Intervals

Abstract

The high-dimensional D. H. Lehmer problem over quarter intervals is studied. By using the properties of character sum and the estimates of Dirichlet -function, the previous result is improved to be the best possible in the case of , an odd prime with (mod 4), which is shown by studying the mean square value of the error term.

1. Introduction and Main Results

Let be an odd integer and let be an integer coprime to . For each integer with and , there is a unique integer with such that . Let denote the number of solutions of the congruence equation with such that are of opposite parity. D. H. Lehmer posed the problem to find or at least to say something nontrivial about it (see problem F12 of [1], page 251), where is an odd prime. Zhang [2] proved that where is the Euler function and is the Dirichlet divisor function. For the further properties of , he [3] studied the mean square value of the error term and obtained For general odd integer , the similar properties of were studied in [4].

It is interesting to study the D. H. Lehmer problem over short intervals with being a real number. Denote by the number of pairs of integers with , , and having different parity. In [5], Xu and Zhang studied the mean square value of error term in the case of and obtained a sharp asymptotic formula

Let be a positive integer and a nonnegative integer; let be real numbers and . Let be a positive integer and an integer coprime to . In [6], Xu and Zhang studied the high-dimensional D. H. Lehmer problem over short intervals as and obtained an interesting asymptotic formula where They also improved the result for in the case to be the best possible, by studying the mean square value of the error term .

In this paper, we consider the high-dimensional D. H. Lehmer problem over quarter intervals. Let in the case . That is, We will use the properties of character sum and the estimates of Dirichlet -function to obtain a sharper asymptotic formula of in the case of , an odd prime with . In order to show that our result is close to be the best possible, the mean square value of is studied too.

In this paper, we will use the following notations:denotes the summation over all primitive characters modulo such that ;  denotes the number of all primitive characters modulo ;  denotes the -th divisor function (i.e., the number of solutions of the equation in positive integers );  denotes the primitive character modulo ;  denotes the primitive character modulo with , and denotes the primitive character modulo with .

The main results are the following.

Theorem 1. Let be an odd prime with and coprime to . Then, for any positive integer with , one has the asymptotic formula that

Theorem 2. Let be an odd prime with . Then, for any positive integer , one has where and where , and is any fixed positive number.

From Theorem 2 we know that for some and thus the bound in Theorem 1 is close to be the best possible.

For general odd number , whether there exist similar asymptotic formulae for in the cases of and are open problems.

2. Several Auxiliary Lemmas

To establish the main results of our theorems, we need the following several auxiliary lemmas.

Lemma 3. Let be an odd integer and let be a primitive Dirichlet character modulo . Then one has

Proof. See [7] or Lemma 2 in [8].

Lemma 4. Let be an odd prime with and coprime to . Then, for any positive integer , one has

Proof. From the definition of we have Note that ; we have Combining the above with Lemma 3, we can get Noting the definition of , we can immediately get Lemma 4.

Lemma 5. Let be an odd integer and coprime to . Then, for any positive integer with , one has

Proof. Using the similar method as proving Lemma 5 of [9], we can obtain these estimates.

Lemma 6. Let and be integers with and , and let be a Dirichlet character modulo . Then one has the identities where denotes the summation over all primitive characters modulo and is the number of primitive characters modulo .

Proof. This is Lemma 3 of [10].

Lemma 7. Let be nonnegative integers with an odd integer, and let be a Dirichlet character modulo . Then, for any positive integer , one has

Proof. Using the similar method as proving Lemma 4 of [11], we can get the results.

Lemma 8. Let be an odd integer, and let be a positive integer. Then, for any fixed nonnegative integers such that , one has where .

Proof. We only prove the first formula; the others can be obtained by the similar method.
For convenience, we put where is a parameter with . Then from Abel's identity we have Hence, we have From the proof of Lemma 6 of [11], we know that only the term which does not contain the infinite integral will make contribution to the main term. That is, Then from Lemma 6, we can write where denotes the summation over all integers with and . Now we split the above first sum into the following four cases:(i) and ;(ii) and ;(iii) and ;(iv) and .
By using the similar method as proving Lemma 6 of [11], we know that the main term will be Hence we have This proves Lemma 8.

Lemma 9. Let be an odd integer and let be a positive integer. Then, for any fixed nonnegative integers such that while , and while , one has

Proof. Using the similar method as proving Lemma 6 of [11], we can get the results.

3. Proof of Theorems

In this section, we will complete the proof of our theorems. From Lemmas 4 and 5, we can immediately get the result of Theorem 1.

Now we come to prove Theorem 2. Noting that if , from Lemma 4 and the orthogonality of Dirichlet character, we can write And consider that Combining the above with Lemmas 7–9, we have where This proves Theorem 2.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The author expresses his gratitude to the referee for his/her very helpful and detailed comments. This work is supported by the NSF (11201275) and the Natural Science Foundation of Shaanxi Province of China (no.2011JQ1010).