On the Distribution of -Tuples of -Free Numbers

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 M？bius 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