##
**Zentralblatt MATH**

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

**Zbl.No: ** 409.10043

**Autor: ** Erdös, Paul; Saffari, B.; Vaughan, R.C.

**Title: ** On the asymptotic density of sets of integers. II. (In English)

**Source: ** J. London Math. Soc., II. Ser. 19, 17-20 (1979).

**Review: ** [Part I, cf. ibid. 13, 475-485 (1976; Zbl 333.10039)]

Let A and B be a pair of direct factors of N^*, the set of positive integers; that is a pair of subsets A and B of N^* such that every n in N^* can be written uniquely as n = a· b, with a in A and b in B. Let S\subset N^* and d(S) denote the asymptotic density of S whenever it exists.Let H(S) = **sum**_{n in s} ^{1}/_{n} . It has been shown by Saffari that in the convergent case, the sets A and B habe asymptotic densities: d(A) = \frac 1{H(B)} and d(B) = \frac 1{H(A)}. In this paper the authors settle (in the affirmative) the first two open problems stated by Saffari. In fact they prove: Theorem 1. The direct factors A and B have asymptotic densities in the divergent case H(A) = H(B) = oo and d(A) = 0. Theorem 2. In the divergent case H(A) = H(B) = oo, we have **sum**_{b in A} ^{1}/_{b} = **sum**_{b in B} ^{1}/_{p} = oo.

**Reviewer: ** D.Suryanarayana

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

**Keywords: ** asymptotic density; sets of integers; direct factors

**Citations: ** Zbl.333.10039

© European Mathematical Society & FIZ Karlsruhe & Springer-Verlag