An improved bound on the least common multiple of polynomial sequences

TL;DR

This paper improves the lower bound on the logarithm of the least common multiple of polynomial sequences, confirming conjectures for irreducible polynomials of degree at least 2 and providing new bounds involving the radical of the lcm.

Contribution

It offers an alternative proof with an improved constant for the lower bound and establishes new bounds involving the radical of the lcm of polynomial sequences.

Findings

Proved $ ext{log lcm} o (d-1)N ext{log}N$ conjecture for irreducible polynomials.

Improved the constant in the lower bound from $rac{d-1}{d^2}$ to 1.

Established a new bound for the radical of the lcm, $ ext{log rad lcm} o rac{2}{d}N ext{log}N$.

Abstract

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