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**Zentralblatt MATH**

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

**Zbl.No: ** 421.10027

**Autor: ** Erdös, Paul; Hall, R.R.

**Title: ** The propinquity of divisors. (In English)

**Source: ** Bull. Lond. Math. Soc. 11, 304-307 (1979).

**Review: ** In [J. London Math. Soc. 39, 692-696 (1964; Zbl 125.08602)], *P.Erdös* stated that, if \beta > log3-1, then the sequence of integers n with two divisors d,d' satisfying d < d' < d(1-(log n)^{-\beta}) has asymptotic density 0. An improvement of this by the present authors shows in effect that one can replace the term (log n)^{-\beta} in the above by 3^{-\lambda(n)}(log d)^{1- log 3} where \lambda(n) = (1-\epsilon)\sqrt{**{**2 log log n· log log log log n**}**} and \epsilon > 0 is fixed. The proof (of an equivalent form of this result) depends on estimating certain weighted sums over the integers n < x of the required type; the case when d > x^{\delta}(\delta > 0) is easier to handle, and indeed a slightly stronger result is derived here.

**Reviewer: ** E.J.Scourfield

**Classif.: ** * 11N05 Distribution of primes

11N37 Asymptotic results on arithmetic functions

11K65 Arithmetic functions (probabilistic number theory)

**Keywords: ** propinquity of divisors; asymptotic density

**Citations: ** Zbl.125.086

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