Normal largest gap between prime factors
Journal de théorie des nombres de Bordeaux, Volume 31 (2019) no. 3, pp. 747-749.

Let {p j (n)} j=1 ω(n) denote the increasing sequence of distinct prime factors of an integer n. We provide details for the proof of a statement of Erdős implying that, for any function ξ(n) tending to infinity with n, we have

f(n):=max1j<ω(n)loglogpj+1(n)logpj(n)=log3n+O(ξ(n))

for almost all integers n.

Désignons par {p j (n)} j=1 ω(n) la suite croissante des facteurs premiers distincts d’un entier n. Nous explicitions les détails de la preuve d’un énoncé d’Erdős impliquant que, pour toute fonction ξ(n) tendant vers l’infini avec n, nous avons

f(n):=max1j<ω(n)loglogpj+1(n)logpj(n)=log3n+O(ξ(n))

pour presque tout entier n.

Received:
Accepted:
Published online:
DOI: 10.5802/jtnb.1107
Classification: 11N56, 14G42
Mots-clés : Distribution of prime factors, normal order, largest gap.

Gérald Tenenbaum 1

1 Institut Élie Cartan Université de Lorraine BP 70239 54506 Vandœuvre-lès-Nancy Cedex, France
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Gérald Tenenbaum. Normal largest gap between prime factors. Journal de théorie des nombres de Bordeaux, Volume 31 (2019) no. 3, pp. 747-749. doi : 10.5802/jtnb.1107. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1107/

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[6] Gérald Tenenbaum Introduction to analytic and probabilistic number theory, Graduate Studies in Mathematics, 163, American Mathematical Society, 2015 | MR | Zbl

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