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On the Normal Growth of Prime Factors of Integers
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- Journal:
- Canadian Journal of Mathematics / Volume 44 / Issue 6 / 01 December 1992
- Published online by Cambridge University Press:
- 20 November 2018, pp. 1121-1154
- Print publication:
- 01 December 1992
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- Article
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Let h: [0,1] → R be such that
and define 
.In 1966, Erdős [8] proved that 
holds for almost all n, which by using a simple argument implies that in the case h(u) = u, for almost all n, 
He further obtained that, for every z > 0 and almost all n, 
and that 
where ϕ, ψ, are continuous distribution functions. Several other results concerning the normal growth of prime factors of integers were obtained by Galambos [10], [11] and by De Koninck and Galambos [6].Let χ = ﹛xm : w ∈ N﹜ be a sequence of real numbers such that limm→∞ xm = +∞. For each x ∈ χ let
be a set of primes p ≤x. Denote by p(n) the smallest prime factor of n. In this paper, we investigate the number of prime divisors p of n, belonging to for which Th(n,p) > z. Given Δ < 1, we study the behaviour of the function 
We also investigate the two functions 
, where, in each case, h belongs to a large class of functions.
and define 
.In 1966, Erdős [8] proved that 
holds for almost all n, which by using a simple argument implies that in the case h(u) = u, for almost all n, 
He further obtained that, for every z > 0 and almost all n, 
and that 
where ϕ, ψ, are continuous distribution functions. Several other results concerning the normal growth of prime factors of integers were obtained by Galambos [10], [11] and by De Koninck and Galambos [6].
be a set of primes p ≤x. Denote by p(n) the smallest prime factor of n. In this paper, we investigate the number of prime divisors p of n, belonging to 
We also investigate the two functions 
, where, in each case, h belongs to a large class of functions.