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On the distribution of square-free numbers. (English) Zbl 0532.10025

Let \(Q_ k(x)\) denote the number of k-free integers not exceeding x, and let \(R_ k(x)\) denote \(Q_ k(x)-x/\zeta(k).\) It is conjectured that \(R_ k(x)<<_{\epsilon,k}\quad x^{1/2k+\epsilon}\) for all \(k\geq 2\) and \(\epsilon>0\), but this is not known, even assuming the Riemann hypothesis. In the opposite direction, C. J. A. Evelyn and E. H. Linfoot [Ann. Math., II. Ser. 32, 261-270 (1931; Zbl 0002.01501)] proved that \(R_ k(x)=\Omega(x^{1/2k})\), and I. Kátai [Acta Arith. 13, 107-122 (1967; Zbl 0159.062)] proved that \[ \int^{y}_{1}x^{-1} R_ k(x) dx\quad>\quad(c/k)\quad Y^{0.36/k}. \] In a characteristically short and ingenious manner, the author sharpens both of these results by proving that \[ Y^{-1}\int^{Y}_{1}| R_ k(x)| \quad dx\quad>\quad c(k)\quad Y^{1/2k}. \]
Reviewer: S.W.Graham

MSC:

11N05 Distribution of primes
11N37 Asymptotic results on arithmetic functions
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