Zentralblatt MATH
Publications of (and about) Paul Erdös
Zbl.No: 272.10021
Autor: Erdös, Paul; Hall, R.R.
Title: Some distribution problems concerning the divisors of integers. (In English)
Source: Acta Arith. 26, 175-188 (1974).
Review: The paper is concerned with the distribution (mod 1) of the numbers log d, where d runs through the divisors of an integer n. Let |x| denote the distance from x to the nearest integer. Then the authors' main result is the following: let \alpha and c be real numbers. The integers n having a divisor d satisfying 0 < | log d- \alpha| < 2- log log n-c \sqrt{log log n} (1) have asymptotic density \delta (c) = (2 \pi)- ½ int ooc e-u2/2du, moreover if c is replaced by a function of n tending to +oo or -oo, the density is 0 or 1 respectively. If there exists an integer m(\alpha), necessarily unique, such that log m(\alpha) \equiv \alpha (mod 1), the density is increased if equality is allowed on the extreme left of (1): it becomes \delta+(1-\delta)/m(\alpha) where \delta = \delta (c).
Classif.: * 11J71 Distribution modulo one
11B83 Special sequences of integers and polynomials
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