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A property of the set of primes as a multiplicative basis of the natural numbers. (English. Russian original) Zbl 1252.11072

Dokl. Math. 84, No. 1, 467-470 (2011); translation from Dokl. Akad. Nauk 439, No. 2, 159-162 (2011).
Let \(k,r,l_1,l_2,\dots, l_r\) be positive integer numbers under conditions: \(k\geq 2\); \(1<r<\varphi(k)\); \(1\leq l_j<k\), (\(l_j,k)=1\) for \(j=1,2,\dots,r\) (here \(\varphi\) denotes the Euler function). Let \(\mathbb{A}\) denote the set of primes in the progressions \(km+l_j,\;j=1,2,\dots, r\). The following asymptotic formula is the crucial result of the paper.
Theorem. Suppose that \(\mathbb{N}^*\) is the set of positive integers \(n\) heaving no primes factors from \(\mathbb{A}\), \(\mathbb{N}_0^*\subset\mathbb{N}^*\) contains those \(n\) which have an even number of prime factors, and \(\mathbb{N}_1^*\subset\mathbb{N}^*\) contains those \(n\) which have an odd number of prime factors. In addition, suppose that: \[ n^*(x)=\sum\limits_{n\leq x,\, n\in\mathbb{N}^* }1,\;\;n_0^*(x)=\sum\limits_{n\leq x,\, n\in\mathbb{N}_0^* }1,\;\;n_1^*(x)=\sum\limits_{n\leq x,\, n\in\mathbb{N}_1^* }1. \] Then \[ n_1^*(x)-n_0^*(x)\sim C\,n^*(x)\left(\log x\right)^{2(r/\varphi(k)-1)},\quad x\rightarrow\infty, \] with some positive absolute constant \(C\).
The similar result is also obtained for square-free numbers.

MSC:

11N37 Asymptotic results on arithmetic functions
11A41 Primes
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References:

[1] E. Landau, Gött. Nachr. 6, 687–771 (1912).
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