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 Qx(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, dk = limx ––> ooQx(x)/x and dk = infn \geq 0Qx(n)/n. H.M.Stark has shown [Proc. Am. Math. Soc. 17, 1211-1214 (1966; Zbl 144.28205)] that dk < Dk for all k > 1, hence that dk = Qk(nk)/nk for at least one integer nk. P.H.Diananda and M.V.Subbarao have proved [Proc. Amer. Math. Soc. 62, 7-10 (1977; Zbl 346.10026)] dk > 1-2-k-3-k-5-k and several related results. It is now proved that

{dk-(1-2-k-3-k-5-k)}//Dk-dk) = o(2/3)k ––> 0 as k ––> oo,

hence that dk is always closer to 1-2-k-3-k-5-k than to Dk. A table of values of nk and Qx(nk), 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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