The least common multiple of a bivariate quadratic sequence
Volume 211 / 2023
Abstract
Let $F\in \mathbb Z[x,y]$ be some polynomial of degree 2. We find the asymptotic behaviour of the least common multiple of the values of $F$ up to $N$. More precisely, we consider $\psi_F(N) = \log(\text{LCM}_{0 \lt F(x,y)\leq N}\lbrace F(x,y) \rbrace)$ as $N$ tends to infinity. It turns out that there are four different possible asymptotic behaviours depending on $F$. For a generic $F$, the function $\psi_F(N)$ has order of magnitude $\frac{N\log\log N}{\sqrt{\log N}}$. This is also 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 determine when each one occurs.
