An improved bound on the least common multiple of polynomial sequences

Ashwin Sah

doi:10.5802/jtnb.1146

Ashwin Sah

Journal de Théorie des Nombres de Bordeaux, Tome 32 (2020) no. 3, pp. 891-899.

Résumé
Abstract

Cilleruelo a conjecturé que si $f \in ℤ [x]$ de degré $d \geq 2$ est irréductible sur les rationnels, alors $log lcm (f (1), ..., f (N)) \sim (d - 1) N log N$ quand $N \to \infty$ . Il l’a prouvé dans le cas $d = 2$ . Très récemment, Maynard et Rudnick ont prouvé qu’il existe $c_{d} > 0$ tel que $log lcm (f (1), ..., f (N)) ≳ c_{d} N log N$ , et ont montré qu’on peut prendre $c_{d} = \frac{d - 1}{d^{2}}$ . Nous donnons une preuve alternative de ce résultat avec la constante améliorée $c_{d} = 1$ . De plus, nous prouvons la minoration $log rad lcm (f (1), ..., f (N)) ≳ \frac{2}{d} N log N$ et proposons une conjecture plus forte affirmant que $log rad lcm (f (1), ..., f (N)) \sim (d - 1) N log N$ quand $N \to \infty$ .

Cilleruelo conjectured that if $f \in ℤ [x]$ of degree $d \geq 2$ is irreducible over the rationals, then $log lcm (f (1), ..., f (N)) \sim (d - 1) N log N$ as $N \to \infty$ . He proved it for the case $d = 2$ . Very recently, Maynard and Rudnick proved there exists $c_{d} > 0$ with $log lcm (f (1), ..., f (N)) ≳ c_{d} N log N$ , and showed one can take $c_{d} = \frac{d - 1}{d^{2}}$ . We give an alternative proof of this result with the improved constant $c_{d} = 1$ . We additionally prove the bound $log rad lcm (f (1), ..., f (N)) ≳ \frac{2}{d} N log N$ and make the stronger conjecture that $log rad lcm (f (1), ..., f (N)) \sim (d - 1) N log N$ as $N \to \infty$ .

Reçu le : 2020-02-22
Révisé le : 2020-09-08
Accepté le : 2020-09-18
Publié le : 2021-01-08

DOI : https://doi.org/10.5802/jtnb.1146
Classification : 11N32
Mots clés : Least common multiple, polynomial sequence

@article{JTNB_2020__32_3_891_0,
     author = {Ashwin Sah},
     title = {An improved bound on the least common multiple of polynomial sequences},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {891--899},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {32},
     number = {3},
     year = {2020},
     doi = {10.5802/jtnb.1146},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1146/}
}

Ashwin Sah. An improved bound on the least common multiple of polynomial sequences. Journal de Théorie des Nombres de Bordeaux, Tome 32 (2020) no. 3, pp. 891-899. doi : 10.5802/jtnb.1146. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1146/

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