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**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 = d_{1} < d_{2} < ... < 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 d_{i}, d_{i+1} of n. If \theta is a real bounded function defined on (0,1) and if F(n; \theta) = **sum**_{1 \leq i < \tau(n)}\theta(d_{1}/d_{i+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) = t^{r}, the result here being uniform for r log x >> 1. The authors also study the sums **sum**_{n \leq x}f(n), **sum**_{n \leq x}g(n) where f(n), g(n) denote the number of pairs d_{i}, d_{i+1} of divisors of n with the property that (d_{i},d_{i+1}) = 1, d_{i}|d_{i+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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