## Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  776.11013
Autor:  Erdös, Paul; Zhang, Zhenxiang
Title:  Upper bound of sum 1/(ai log ai) for primitive sequences. (In English)
Source:  Proc. Am. Math. Soc. 117, No.4, 891-895 (1993).
Review:  A sequence A = {ai} of positive integers a1 < a2 < ··· is called primitive if ai\nmid aj for i\ne j. Define f(A) = suma in A (1/a log a). In 1935, the first author proved that there exists an absolute constant c such that f(A) < c for any primitive sequence A. The main result of this paper is that c = 1.84 is admissible. The authors also give a necessary and sufficient condition for a more recent conjecture of the first author namely that for any primitive sequence A,

suma \leq n,a in A{1\over a log a} \leq sump \leq n{1\over p log p}   (n > 1),

where p denotes a prime number.
Reviewer:  M.Nair (Glasgow)
Classif.:  * 11B83 Special sequences of integers and polynomials
Keywords:  upper bound; primitive sequence

© European Mathematical Society & FIZ Karlsruhe & Springer-Verlag