## Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  174.08104
Autor:  Erdös, Pál
Title:  On the distribution of prime divisors (In English)
Source:  Aequationes Math. 2, 177-183 (1969).
Review:  Denote by v(n; a,b) the number of distinct prime factors of n satisfying a \leq p \leq b. v(n) = v(n; l,n) denotes the number of distinct prime factors of n. A well known theorem of Hardy and Ramanujan states that for almost all n,v(n) = (1+o(1)) log log n. The principal aim of this paper is to prove that if b-a/ log log n ––> oo then for almost all integers n

v(n; a,b) = (1+o(1))(log log b- log log a)

uniformly in a and b. More precisely to every \epsilon > 0 there is a c so that if we neglect o(x) integers n < x then for every a,b satisfying \TagsOnLeft

log log b- log log a > c log log log n    (1)

we have

(1-\epsilon)(log log b- log log a) < v(n; a,b) < (1+\epsilon)(log log b - log log a).

We also show that (1) is essentially best possible. The proof uses Turán's method and other ideas of probabilistic number theory. Some related results in probability and number theory are also discussed.
Classif.:  * 11N25 Distribution of integers with specified multiplicative constraints
11N30 Turan theory
Index Words:  number theory

© European Mathematical Society & FIZ Karlsruhe & Springer-Verlag