The Least Common Multiple of a Bivariate Quadratic Sequence

Abstract

Let $F\in\mathbb{Z}[x,y]$ be some polynomial of degree 2. In this paper we find the asymptotic behaviour of the least common multiple of the values of $F$ up to $N$. More precisely, we consider $ψ_F(N) = \log\left(\text{LCM}_{0<F(x,y)\leq N}\left\lbrace F(x,y)\right\rbrace\right)$ as $N$ tends to infinity. It turns out that there are 4 different possible asymptotic behaviours depending on $F$. For a generic $F$, we show that the function $ψ_F(N)$ has order of magnitude $\frac{N\log\log N}{\sqrt{\log N}}$. We also show that this is the expected order of magnitude according to a suitable random model. However, special polynomials $F$ can have different behaviours, which sometimes deviate from the random model. We give a complete description of the order of magnitude of these possible behaviours, and when each one occurs.