On the Normal Growth of Prime Factors of Integers

Koninck, J. M. De ; Kátai, I. ; Mercier, A.

Canadian journal of mathematics, Tome 44 (1992) no. 6, pp. 1121-1154

Résumé

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.

Détail

DOI : 10.4153/CJM-1992-068-6

Mots-clés : 11K65, 11N25, 11N35, prime factors, distribution functions, continuity module

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