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Erdős, Pál

On the distribution of prime divisors. (English) Zbl 0174.08104

Aequationes Math. 2, 177-183 (1969).
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 \(b-a/\log\log n \to \infty\) 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 \(\varepsilon >0\) there is a \(c\) so that if we neglect \(o(x)\) integers \(n<x\) then for every \(a,b\) satisfying \[ \log\log b-\log\log a>c\log\log\log n\tag{1} \] we have \[ (1-\varepsilon)(\log\log b-\log\log a)<v(n;a,b)<(1+\varepsilon)(\log\log b -\log\log a). \] 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.
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Cited in 3 Reviews
Cited in 7 Documents

MSC:

11N25 Distribution of integers with specified multiplicative constraints
11N30 Turán theory

Keywords:

number theory
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\textit{P. Erdős}, Aequationes Math. 2, 177--183 (1969; Zbl 0174.08104)
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References:

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[7] See, e.g.Halberstam, H., Roth, K.F.,Sequences, Vol. I (Clarendon Press, Oxford 1966), Chapt. IV.
[8] Hardy, G.H. andRamanujan, S.,The normal number of prime factors of n, Quart. J. Math.48, 76–92 (1917). · JFM 46.0262.03
[9] Kubylyus, I.P.,Probabilistic methods in number theory, Amer. Math. Soc. Transl. [2]19, 47–85 (1962).
[10] Levy, P.,Théorie de l’addition des variables aléatoires (Gauthier-Villars, Paris 1937).
[11] Turán, P.,On a theorem of Hardy and Ramanujan, J. London Math. Soc.9, 274–276 (1934). · Zbl 0010.10401
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