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**Zentralblatt MATH**

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

**Zbl.No: ** 627.10035

**Autor: ** Erdös, Paul; Lacampagne, Carole B.; Pomerance, Carl; Selfridge, J.L.

**Title: ** On the Schnirelmann and asymptotic densities of sets of non-multiples. (In English)

**Source: ** Combinatorics, graph theory and computing, Proc. 16th Southeast. Conf., Boca Raton/Fla. 1985, Congr. Numerantium 48, 67-79 (1985).

**Review: ** [For the entire collection see Zbl 619.00006.]

Let \delta(S), \sigma(S) denote the asymptotic density (when it exists), the Schnirelmann density, respectively, of the set of natural numbers not divisible by any element of a set S of natural numbers, and let D(S) = \delta(S)-\sigma(S) \geq 0. When S is a finite set or a subset of the set *P* of all primes, the authors prove some interesting results concerning D(S); for example: (1) \sup **{**D(S): S finite**}** = 1. (2) If S\subset *P*, there exists S' with S\subset S'\subset *P* such that \sigma(S') = \sigma(S) and D(S') = 0.

They also derive upper and lower bounds for \sup **{**D(S): S\subset *P***}**. The paper concludes with a stimulating discussion describing related unsolved problems, their setting and implications.

**Reviewer: ** E.J.Scourfield

**Classif.: ** * 11B05 Topology etc. of sets of numbers

11N37 Asymptotic results on arithmetic functions

11A25 Arithmetic functions, etc.

**Keywords: ** sets of non-multiples; asymptotic density; Schnirelmann density; finite set; upper and lower bounds; unsolved problems

**Citations: ** Zbl 619.00006

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