Normal largest gap between prime factors

Tenenbaum, Gérald

Journal de théorie des nombres de Bordeaux, Tome 31 (2019) no. 3, pp. 747-749.

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 $n$.

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 $n$.

Classification : 11N56, 14G42
Mots clés : Distribution of prime factors, normal order, largest gap.

[1] Erdős, Pál Some remarks on prime factors of integers, Can. J. Math., Volume 11 (1959), pp. 161-167 | DOI | MR | Zbl

[2] Erdős, Pál On some properties of prime factors of integers, Nagoya Math. J., Volume 27 (1966), pp. 617-623 | DOI | MR | Zbl

[3] Erdős, Pál On the distribution of prime divisors, Aequationes Math., Volume 2 (1969), pp. 177-183 | DOI | MR | Zbl

[4] Sofos, Efthymios (private e-mail message, August 30, 2018)

[5] Tenenbaum, Gérald Cribler les entiers sans grand facteur premier, Philos. Trans. R. Soc. Lond., Ser. A, Volume 345 (1993), pp. 377-384 | MR | Zbl

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