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.

Cilleruelo a conjecturé que si f[x] de degré d2 est irréductible sur les rationnels, alors loglcm(f(1),...,f(N))(d-1)NlogN quand N. Il l’a prouvé dans le cas d=2. Très récemment, Maynard et Rudnick ont prouvé qu’il existe c d >0 tel que loglcm(f(1),...,f(N))c d NlogN, et ont montré qu’on peut prendre c d =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 logradlcm(f(1),...,f(N))2 dNlogN et proposons une conjecture plus forte affirmant que logradlcm(f(1),...,f(N))(d-1)NlogN quand N.

Cilleruelo conjectured that if f[x] of degree d2 is irreducible over the rationals, then loglcm(f(1),...,f(N))(d-1)NlogN as N. He proved it for the case d=2. Very recently, Maynard and Rudnick proved there exists c d >0 with loglcm(f(1),...,f(N))c d NlogN, and showed one can take c d =d-1 d 2 . We give an alternative proof of this result with the improved constant c d =1. We additionally prove the bound logradlcm(f(1),...,f(N))2 dNlogN and make the stronger conjecture that logradlcm(f(1),...,f(N))(d-1)NlogN as N.

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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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