Zentralblatt MATH
Publications of (and about) Paul Erdös
Zbl.No: 526.10036
Autor: Erdös, Paul; Tenenbaum, G.
Title: Sur les diviseurs consecutifs d'un entier. (On consecutive divisors of an integer.) (In French)
Source: Bull. Soc. Math. Fr. 111, 125-145 (1983).
Review: Let 1 = d1 < d2 < ... < d\tau(n) = n denotes the divisors of n. The well known conjecture of the first author that almost all positive integers n have a pair d, d' of divisors such that d < d' \leq 2d has prompted various investigations into the behaviour of pairs of divisors of an integer.
In this interesting paper; the authors consider some properties of pairs of consecutive divisors di, di+1 of n. If \theta is a real bounded function defined on (0,1) and if F(n; \theta) = sum1 \leq i < \tau(n)\theta(d1/di+1), it is established in Theorem 1 that F(n; \theta)/\tau(n) has a distribution function. An asymptotic formula for sum{n \leq x}F(n; \theta) is derived in Theorem 2 for a class of functions \theta and in Theorem 3 for the function given by \theta(t) = tr, the result here being uniform for r log x >> 1. The authors also study the sums sumn \leq xf(n), sumn \leq xg(n) where f(n), g(n) denote the number of pairs di, di+1 of divisors of n with the property that (di,di+1) = 1, di|di+1, respectively. By choosing the function \theta appropriately, it follows from Theorem 1 that g(n)/\tau(n) has a distribution function, as conjectured by the authors in [Ann. Inst. Fourier 31, No. 1, 17-37 (1981; Zbl 437.10020)]. The proofs in this paper depend on some rather intricate handling of various double sums involving the characteristic functions associated with certain divisibility properties, and are rather complicated.
Reviewer: E.J.Scourfield
Classif.: * 11N37 Asymptotic results on arithmetic functions
11K65 Arithmetic functions (probabilistic number theory)
Keywords: arithmetical function; pairs of consecutive divisors; asymptotic formula; distribution function; number of pairs of divisors
Citations: Zbl.456.10022; Zbl.437.10020
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