Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  146.27101
Autor:  Erdös, Pál; Sarközy, A.; Szemeredi, E.
Title:  On a theorem of Behrend (In English)
Source:  J. Aust. Math. Soc. 7, 9-16 (1967).
Review:  A sequence A = {ai} of positive, increasing integers is called primitive if no term divides another. Let log2 x = log log x and fA(x) = sumai < x ai-1 and denote by ci absolute, positive constants. F.Behrend (Zbl 012.05203) proved: If A is primitive, then there exists c1 such that

fA(x) < c1 log x (log2 x) ½.

Pillai showed that there exists c2 such that for every x there is a finite primitive sequence A, with ai \leq x for which fA(x) > c2 log x (log2 x)- ½, so that Behrend's theorem is, in this sense, best possible.
In the present paper it is shown that if A is infinite, Behrend's result can be improved to read

(^*) fA(x) = o(log x (log2 x)- ½).

However, no further improvement is possible, because, if h(x) ––> oo arbitrarily slowly, then there exists a primitive sequence A, so that

limsupx ––> oo fA(x) h(x) (log2 x) ½ (log x)-1 = oo.

For a proof, the general case is first reduced to that of squarefree ai's; then the author use a lemma, which, while technically elementary, has a rather long and difficult proof. A sharper form of (^*) and some related topics are also discussed and proved.
Reviewer:  E.Grosswald
Classif.:  * 11B83 Special sequences of integers and polynomials
Index Words:  number theory

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