Normal largest gap between prime factors
Journal de Théorie des Nombres de Bordeaux, Tome 31 (2019) no. 3, pp. 747-749.

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.

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.

Reçu le :
Accepté le :
Publié le :
DOI : https://doi.org/10.5802/jtnb.1107
Classification : 11N56,  14G42
Mots clés : Distribution of prime factors, normal order, largest gap.
@article{JTNB_2019__31_3_747_0,
     author = {Tenenbaum, G\'erald},
     title = {Normal largest gap between prime factors},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {747--749},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {31},
     number = {3},
     year = {2019},
     doi = {10.5802/jtnb.1107},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/jtnb.1107/}
}
TY  - JOUR
AU  - Tenenbaum, Gérald
TI  - Normal largest gap between prime factors
JO  - Journal de Théorie des Nombres de Bordeaux
PY  - 2019
DA  - 2019///
SP  - 747
EP  - 749
VL  - 31
IS  - 3
PB  - Société Arithmétique de Bordeaux
UR  - http://www.numdam.org/articles/10.5802/jtnb.1107/
UR  - https://doi.org/10.5802/jtnb.1107
DO  - 10.5802/jtnb.1107
LA  - en
ID  - JTNB_2019__31_3_747_0
ER  - 
Tenenbaum, Gérald. Normal largest gap between prime factors. Journal de Théorie des Nombres de Bordeaux, Tome 31 (2019) no. 3, pp. 747-749. doi : 10.5802/jtnb.1107. http://www.numdam.org/articles/10.5802/jtnb.1107/

[1] Erdős, Pál Some remarks on prime factors of integers, Can. J. Math., Volume 11 (1959), pp. 161-167 | Article | MR 104634 | Zbl 0092.04301

[2] Erdős, Pál On some properties of prime factors of integers, Nagoya Math. J., Volume 27 (1966), pp. 617-623 | Article | MR 204378 | Zbl 0151.03501

[3] Erdős, Pál On the distribution of prime divisors, Aequationes Math., Volume 2 (1969), pp. 177-183 | Article | MR 244178 | Zbl 0174.08104

[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 1253499 | Zbl 0795.11042

[6] Tenenbaum, Gérald Introduction to analytic and probabilistic number theory, Graduate Studies in Mathematics, 163, American Mathematical Society, 2015 | MR 3363366 | Zbl 1336.11001

Cité par Sources :