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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