## Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  015.10001
Autor:  Davenport, H.; Erdös, Pál
Title:  On sequences of positive integers. (In English)
Source:  Acta Arith. 2, 147-151 (1936).
Review:  Let a1,a2,... be any sequence of different positive integers, and b1,b2,... the integers divisible by at least on a. It was proved by A.S.Besicovitch (Zbl 009.39504) that the sequence {bi} may have different upper and lower densities. Here it is shown that the logarithmic density

limx ––> oo (log x)-1 sumbi \leq x bi-1

exists and is equal to the lower density of the sequence. The proof uses Dirichlet series. It is deduced that if a sequence a1,a2,... has a positive upper logarithmic density, then it has a subsequence ai1,ai2,... in which aik | ai_{k+1} (k = 1,2,...).
Reviewer:  E.C.Titchmarsh (Oxford)
Classif.:  * 11B83 Special sequences of integers and polynomials
11B05 Topology etc. of sets of numbers
Index Words:  Algebra, number theory

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