Abstract
For k, being a fixed integer ≥2, a positive integer n is called k-free number if n is not divisible by the kth power of any integer >1. In this paper, we studied the distribution of r-tuples of k-free numbers and derived an asymptotic formula.
1. Introduction
A positive integer is called square-free number if it is not divisible by a perfect square except . Let be the characteristic function of the sequence of square-free numbers. That is,
From [1] we know that
Mirsky [2] studied the frequency of pairs of square-free numbers with a given difference and proved the asymptotic formula:
Heath-Brown [3] investigated the number of consecutive square-free numbers not more than and obtained the following result:
Pillai [4] gave an asymptotic formula for
Tsang [5] proved the following.
Proposition 1. Let be distinct integers with and
For we have
where is the number of distinct residue classes moduli represented by the numbers .
For , being a fixed integer ≥2, a positive integer is called -free number if is not divisible by the th power of any integer >1. Let be the characteristic function of the sequence of -free integers. Gegenbauer [6] proved that
Mirsky [7] showed that and in [2] Mirsky improved the error term to
Meng [8] further improved this result as follows:
Moreover, some recent results on pairs of -free numbers are given in [9, 10].
In this paper, we will study the distribution of -tuples of -free numbers by using the Buchstab-Rosser sieve and the methods in [5]. Our main result is the following.
Theorem 2. Let be distinct integers with and
For we have
where is the number of distinct residue classes moduli represented by the numbers .
Remark 3. Taking in Theorem 2, we immediately get Proposition 1.
The main tool in our argument is the Buchstab-Rosser sieve. Let be a sequence of positive numbers which lie between and and let be a real number. Define . For any square-free number with , , let , where is the Möbius function and when the following set of inequalities is satisfied. Otherwise, .
Similarly, let , where when the following set of inequalities is satisfied. Otherwise, .
From [5] we know that for any positive integer and for any positive integer whose smallest prime factor does not exceed .
In our case we take . Throughout this paper is an unspecified absolute constant.
2. Proof of Theorem 2
To prove the theorem we need the following lemma.
Lemma 4. For any and any positive integers , , there exist absolute constants and such that
Proof. This lemma can be proved by using the methods of the lemma in [5] with a slight modification. For completeness we give a proof.
From the result of Ramanujan [11]
and the observation we can obtain (19).
For any , the inequalities and imply
where is an absolute constant. Now we define
If , we have . While if , we have .
Without loss of generality, we assume that . Then
This ends the proof of (20).
New we prove Theorem 2. Define , , for . Then
Let ; then we have
Define ; then any that is not -free has a divisor ≤. By (18) we have where
It is easy to show that the congruence has solutions modulo . Therefore the congruence has solutions modulo , by the Chinese remainder theorem. Then we have
Therefore
Let
By (16) we have where
It is easy to show that
On the other hand, the inequality implies
Then from (20) we have
Now combining (32)–(36) we get
Furthermore, by (16), we have
From (13) of [5] we know that
And by (19) we get
Therefore
Now from (30), (37), and (41) we immediately get
Taking we have
since
Choosing from (42) and (44), we immediately deduce that
This proves (13).
Using the similar methods we have where
Moreover, by using the argument leading to (41), we can have
Therefore
Taking we have
Therefore
This completes the proof of (14).
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work is supported by the National Natural Science Foundation of China under Grant no. 11201370, the Natural Science Foundation of Shaanxi Province of China under Grant nos. 2013JM1017 and 2011JQ1010, and the Natural Science Foundation of the Education Department of Shaanxi Province of China under Grant no. 2013JK0558.