An improved bound on the least common multiple of polynomial sequences

Sah, Ashwin

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

Cilleruelo conjectured that if $f\in \mathbb{Z}\left[x\right]$ of degree $d\ge 2$ is irreducible over the rationals, then $log\mathrm{lcm}\left(f\right(1),...,f(N\left)\right)\sim (d-1)NlogN$ as $N\to \infty$. He proved it for the case $d=2$. Very recently, Maynard and Rudnick proved there exists $c_d>0$ with $log\mathrm{lcm}(f\left(1\right),...,f\left(N\right))\gtrsim c_{d}NlogN$, 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 $log\mathrm{rad}\mathrm{lcm}(f\left(1\right),...,f\left(N\right))\gtrsim \frac{2}{d}NlogN$ and make the stronger conjecture that $log\mathrm{rad}\mathrm{lcm}\left(f\right(1),...,f(N\left)\right)\sim (d-1)NlogN$ as $N\to \infty$.

Cilleruelo a conjecturé que si $f\in \mathbb{Z}\left[x\right]$ de degré $d\ge 2$ est irréductible sur les rationnels, alors $log\mathrm{lcm}\left(f\right(1),...,f(N\left)\right)\sim (d-1)NlogN$ 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 $log\mathrm{lcm}(f\left(1\right),...,f\left(N\right))\gtrsim c_{d}NlogN$, 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 $log\mathrm{rad}\mathrm{lcm}(f\left(1\right),...,f\left(N\right))\gtrsim \frac{2}{d}NlogN$ et proposons une conjecture plus forte affirmant que $log\mathrm{rad}\mathrm{lcm}\left(f\right(1),...,f(N\left)\right)\sim (d-1)NlogN$ quand $N\to \infty$.

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

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