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**Zentralblatt MATH**

**Publications of (and about) Paul Erdös**

**Zbl.No: ** 417.10047

**Autor: ** Erdös, Paul; Hardy, G.E.; Subbarao, M.V.

**Title: ** On the Schnirelmann density of k-free integers. (In English)

**Source: ** Indian J. Math. 20, 45-56 (1978).

**Review: ** For each integer k \geq 1 let Q_{x}(x) be the number of k-free integers not exceeding x, and let the asymptotic and Schnirelmann densities of the set of k-free integers be, respectively, d_{k} = **lim**_{x ––> oo}Q_{x}(x)/x and d_{k} = **inf**_{n \geq 0}Q_{x}(n)/n. *H.M.Stark* has shown [Proc. Am. Math. Soc. 17, 1211-1214 (1966; Zbl 144.28205)] that d_{k} < D_{k} for all k > 1, hence that d_{k} = Q_{k}(n_{k})/n_{k} for at least one integer n_{k}. *P.H.Diananda* and *M.V.Subbarao* have proved [Proc. Amer. Math. Soc. 62, 7-10 (1977; Zbl 346.10026)] d_{k} > 1-2^{-k}-3^{-k}-5^{-k} and several related results. It is now proved that **{**d_{k}-(1-2^{-k}-3^{-k}-5^{-k})**}**//D_{k}-d_{k}) = o(2/3)^{k} ––> 0 as k ––> oo, hence that d_{k} is always closer to 1-2^{-k}-3^{-k}-5^{-k} than to D_{k}. A table of values of n_{k} and Q_{x}(n_{k}), 1 \leq k \leq 75 is included in the paper, and several conjectures and problems are put forth.

**Reviewer: ** B.Garrison

**Classif.: ** * 11B83 Special sequences of integers and polynomials

11N05 Distribution of primes

**Keywords: ** k-free integers; asymptotic density; Schnirelmann density

**Citations: ** Zbl.144.282; Zbl.346.10026

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