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

Let ${\left\{{p}_{j}\left(n\right)\right\}}_{j=1}^{\omega \left(n\right)}$ 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 $\xi \left(n\right)$ tending to infinity with $n$, we have

 $f\left(n\right):=\underset{1⩽j<\omega \left(n\right)}{max}log\left(\frac{log{p}_{j+1}\left(n\right)}{log{p}_{j}\left(n\right)}\right)={log}_{3}n+O\left(\xi \left(n\right)\right)$

for almost all integers $n$.

Désignons par ${\left\{{p}_{j}\left(n\right)\right\}}_{j=1}^{\omega \left(n\right)}$ 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 $\xi \left(n\right)$ tendant vers l’infini avec $n$, nous avons

 $f\left(n\right):=\underset{1⩽j<\omega \left(n\right)}{max}log\left(\frac{log{p}_{j+1}\left(n\right)}{log{p}_{j}\left(n\right)}\right)={log}_{3}n+O\left(\xi \left(n\right)\right)$

pour presque tout entier $n$.

Accepted:
Published online:
DOI: 10.5802/jtnb.1107
Classification: 11N56,  14G42
Keywords: Distribution of prime factors, normal order, largest gap.
Tenenbaum, Gérald 1

1 Institut Élie Cartan Université de Lorraine BP 70239 54506 Vandœuvre-lès-Nancy Cedex, France
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Tenenbaum, Gérald. 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. http://archive.numdam.org/articles/10.5802/jtnb.1107/

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