An improved bound on the least common multiple of polynomial sequences

2019/11/01 by Sah, Ashwin
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1911.00168

Abstract

Cilleruelo conjectured that if f∈ℤ[x] of degree d≥ 2 is irreducible over the rationals, then \loglcm(f(1),…,f(N))∼(d-1)Nlog N 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))\gtrsim c_d Nlog N, and showed one can take c_d = (d-1)/(d²). 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))\gtrsim(2)/(d)Nlog N and make the stronger conjecture that \logradlcm(f(1),…,f(N))∼ (d-1)Nlog N as N→∞.

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