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**Zentralblatt MATH**

**Publications of (and about) Paul Erdös**

**Zbl.No: ** 723.11046

**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} **sum**_{n \leq x}f(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 \Omega_{z}(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 |\Omega_{z}(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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