An improved bound on the least common multiple of polynomial sequences

Abstract

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