An improved bound on the least common multiple of polynomial sequences

Ashwin Sah

doi:10.5802/jtnb.1146

Ashwin Sah ¹

¹ Department 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
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 $N \to \infty$ .

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

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     journal = {Journal de th\'eorie des nombres de Bordeaux},
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     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
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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. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1146/

References
Cited by

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[2] Javier Cilleruelo The least common multiple of a quadratic sequence, Compos. Math., Volume 147 (2011) no. 4, pp. 1129-1150 | DOI | MR | Zbl

[3] Shaofang Hong; Yuanyuan Luo; Guoyou Qian; Chunlin Wang Uniform lower bound for the least common multiple of a polynomial sequence, C. R. Math. Acad. Sci. Paris, Volume 351 (2013) no. 21-22, pp. 781-785 | DOI | MR | Zbl

[4] Shaofang Hong; Guoyou Qian; Qianrong Tan The least common multiple of a sequence of products of linear polynomials, Acta Math. Hung., Volume 135 (2012) no. 1-2, pp. 160-167 | DOI | MR | Zbl

[5] James Maynard; Ze’ev Rudnick A lower bound on the least common multiple of polynomial sequences (to appear in Riv. Mat. Univ. Parma)

[6] Trygve Nagel Généralisation d’un théorème de Tchebycheff, Journ. de Math., Volume 8 (1921) no. 4, pp. 343-356 | Zbl

[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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