Publications of (and about) Paul Erdös
Autor: Erdös, Paul; Pomerance, Carl
Title: On a theorem of Besicovitch: Values of arithmetic functions that divide their arguments. (In English)
Source: Indian J. Math. 32, No.3, 279-287 (1990).
Review: The authors' starting point is the following result of C. N. Cooper and R. E. Kennedy: Given an arithmetic function f: N > N, the set of integers n with the property f(n)|n has asymptotic density zero, if \mu(x) = x-1 sumn \leq xf(n) > oo and \sigma(x)/\mu(x) > 0, where \sigma (x) denotes the standard deviation.
The authors prove (Theorem 1) the same conclusion in a rather simple way, if f has normal order g, with g\nearrow oo and g(n)/n > 0.
Next they show (Theorem 2) that, for a function g satisfying the conditions given above, the set of integers n with a divisor in the interval (g(n),2g(n)] has asymptotic density zero.
The following lemma, used in the proof of Theorem 2, seems to be of independent interest: Denote by \Omegaz(n) the number of prime and prime-power factors of n not exceeding z. If z \geq 3, and 0 < C < D, then the number of integers n in the interval (C,D] with the property |\Omegaz(n)- log log z| \geq 1/3· log log z may be uniformly estimated by << (D-C)· (log log z)-1+z· (log z· log log z)-1.
Reviewer: W.Schwarz (Frankfurt / Main)
Classif.: * 11N37 Asymptotic results on arithmetic functions
11N25 Distribution of integers with specified multiplicative constraints
Keywords: values of arithmetical functions; integers with divisors in some prescribed intervals; theorem of Cooper and Kennedy; Turán-Kubilius inequality; asymptotic density zero
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