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**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 = **{**a_{i}**}** of positive, increasing integers is called primitive if no term divides another. Let log_{2} x = log log x and f_{A}(x) = **sum**_{ai < x} a_{i}^{-1} and denote by c_{i} absolute, positive constants. *F.Behrend* (Zbl 012.05203) proved: If A is primitive, then there exists c_{1} such that f_{A}(x) < c_{1} log x (log_{2} x)^{ ½}. *Pillai* showed that there exists c_{2} such that for every x there is a finite primitive sequence A, with a_{i} \leq x for which f_{A}(x) > c_{2} log x (log_{2} 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

(^*) f_{A}(x) = o(log x (log_{2} x)^{- ½}). However, no further improvement is possible, because, if h(x) ––> oo arbitrarily slowly, then there exists a primitive sequence A, so that

**limsup**_{x ––> oo} f_{A}(x) h(x) (log_{2} x)^{ ½} (log x)^{-1} = oo. For a proof, the general case is first reduced to that of squarefree a_{i}'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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