## Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  146.27102
Autor:  Erdös, Pál; Sarközy, A.; Szemeredi, E.
Title:  On the divisibility properties of sequences of integers. I (In English)
Source:  Acta Arith. 11, 411-418 (1966).
Review:  Let A = {an} be a sequence of integers; set f(x) = sumai | aj,aj \leq x 1. The main result of this paper is Theorem 1. If A has positive upper logarithmic density c1, then there exists c2, depending on c1 only, so that for infinitely many x, f(x) > x\exp{c2(log2 x) ½ log3 x}. On the other hand, there exists a sequence A of positive upper logarithmic density c1, so that for all x,f(x) < x\exp{c3(log2 x) ½ log3 x}. (All ci stand for positive constants and logkx for the iterated logarithm.) The first inequality is proved using a purely combinatorial Theorem: Let S be a set of n elements and let B1,...,B2, z > c42n (0 < c4 < 1) be subsets of S. Then, if n > n0(c4), one of B's contains at least \exp(c5 n ½ log n) of the B's, where c5 depends only on c4. The second inequality is proved using a result of probabilistic number theory.
Reviewer:  E.Grosswald
Classif.:  * 11B83 Special sequences of integers and polynomials
11B05 Topology etc. of sets of numbers
Index Words:  number theory

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