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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