On the Normal Growth of Prime Factors of Integers

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Abstract
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.

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