Zentralblatt MATH

Publications of (and about) Paul Erdös
Zbl.No:  489.10041
Autor:  Erdös, Paul; Lint, J.H.van
Title:  On the average ratio of the smallest and largest prime divisor of n. (In English)
Source:  Indag. Math. 44, 127-132 (1982).
Review:  Let p(n) and P(n) denote respectively the smallest and the largest prime divisor of the integer n, n \geq 2. In 1974, J. van de Lune showed that sumn \leq x\frac{p(n)}{P(n)} = o(x), x ––> oo. In this paper the authors sharpen this to

sumn \leq x\frac{p(n)}{P(n)} = \frac{x}{log x}+\frac{3x}{(log x)2}(1+o(1)),x ––> oo.

In a first section they prove the simpler estimate \pi(x)(1+o(1)) for the sum by fairly elemtary methods: splitting the positive integers n up tp x into three classes: (i) the primes and primes powers, (ii) the numbers with two different prime factors, (iii) the rest. These classes are in turn subdivided and of each subclass the contribution to the sum is estimated. The main term is of course due to the primes. In a second section the subdivisions and applied methods of estimating the varous sums are much more refined. The primes contribute, according to the prime number theorem \frac{x}{log x}+\frac{x}{log x2}(1+o(1)). The integers of the form pq, p, q prime, p\neq q, give a contribution (2x/(log x)2)(1+o(1)) and all the rest goes into the remainder term. Although rather complicated, the proof is clearly presented and therfor easy to follow.
Reviewer:  H.Jager
Classif.:  * 11N05 Distribution of primes
                   11N37 Asymptotic results on arithmetic functions
                   11A41 Elemementary prime number theory
Keywords:  average ratio; smallest prime factor; largest prime factor; asymptotic behavior; remainder term

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