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
However, no further improvement is possible, because, if h(x) > oo arbitrarily slowly, then there exists a primitive sequence A, so that
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.
Classif.: * 11B83 Special sequences of integers and polynomials
Index Words: number theory
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