Publications of Paul Erdos

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 log₂ x = log log x and f_A(x) = sum_{a_i < 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 c₁ such that

f_A(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 a_i \leq x for which f_A(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

(^*) f_A(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

limsup_{x ––> oo} f_A(x) h(x) (log₂ 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

Books Problems Set Theory Combinatorics Extremal Probl/Ramsey Th.

Graph Theory Add.Number Theory Mult.Number Theory Analysis Geometry

Probabability Personalia About Paul Erdös Publication Year Home Page

Books	Problems	Set Theory	Combinatorics	Extremal Probl/Ramsey Th.
Graph Theory	Add.Number Theory	Mult.Number Theory	Analysis	Geometry
Probabability	Personalia	About Paul Erdös	Publication Year	Home Page