Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  476.10045
Autor:  Erdös, Pál; Szemeredi, Endre
Title:  Remarks on a problem of the American Mathematical Monthly. (In Hungarian)
Source:  Mat. Lapok 28, 121-124 (1980).
Review:  Let A = a1 < a2 < ... be a sequence of positive integers. Let F(A,x,i) denote the number of k's for which the least common multiple [ak,ak+1,...,ak+i-1] satisfies in the inequality [ak,ak+1,...,ak+i-1] \leq x. Some years ago P.Erdös formulated the problem in Am. Math. Mon. to prove that F(A,x,i) < ci x1/i, where ci is a constant depending only on i. This statement is false. This can be seen from the following results of the paper (see III).
I. For any A we have

lim\supx ––> oo \frac{F(A,x,2)}{\sqrt{x}} \leq sumk = 1oo \frac{k ½-(k-1) ½}{k}


liminfx ––> oo \frac{F(A,x,2)}{\sqrt{x}} = 0    (1)

provided that in (1) the sign = holds.
II. For any A we have liminfx ––> oo \frac{F(A,x,2)}{\sqrt{x}} \leq ½ .
III. If i > 4, then there exists an \alphai > 0 such that for each sufficiently large x and suitable A we have

F(A,x,i) > x 1/i +\alphai.

The authors conjecture that III holds for i = 4, too. For i = 3 they have proved that for each sufficiently large x and any A F(A,x,3) < c0x1/3 log x (c0 > 0) and that there is such an A that F(A,x,3) > c1x1/3 log x (c1 > 0) for infinitely many x holds. The question whether there is such an A that for each x the inequality F(A,x,3) > c2x1/3 log x (c2) holds, remain open.
Reviewer:  T.Salát
Classif.:  * 11B83 Special sequences of integers and polynomials
                   11A05 Multiplicative structure of the integers
Keywords:  sequence of positive integers; least common multiple

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