An improved bound on the least common multiple of polynomial sequences

Ashwin Sah

doi:10.5802/jtnb.1146

Ashwin Sah ¹

¹Department of Mathematics Massachusetts Institute of Technology Cambridge, MA 02139, USA

Journal de théorie des nombres de Bordeaux, Volume 32 (2020) no. 3, pp. 891-899

Abstract (Original language)
Résumé

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 $N \to \infty$ .

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 $N \to \infty$ .

Article information
Cite this article

Received: 2020-02-22
Revised: 2020-09-08
Accepted: 2020-09-18
Published online: 2021-01-08

DOI: 10.5802/jtnb.1146

Classification: 11N32
Keywords: Least common multiple, polynomial sequence

Author's affiliations:

Ashwin Sah ¹

¹ Department of Mathematics Massachusetts Institute of Technology Cambridge, MA 02139, USA

License:

CC-BY-ND 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

Ashwin Sah. An improved bound on the least common multiple of polynomial sequences. Journal de théorie des nombres de Bordeaux, Volume 32 (2020) no. 3, pp. 891-899. doi: 10.5802/jtnb.1146

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References
Cited by

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[7] Ze’ev Rudnick; Sa’ar Zehavi On Cilleruelo’s conjecture for the least common multiple of polynomial sequences (2019) (https://arxiv.org/abs/1902.01102)

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