On the Asymptotic Density of a GCD-based Map

Abstract

We show that the symmetry of \[f\left(a,b\right)=\frac{\operatorname{gcd}\left(ab,a+b\right)}{\operatorname{gcd}\left(a,b\right)}\] stems from an ${\mathrm{SL}}_2\left(\mathbb{Z}\right)$ action on primitive pairs and that all solutions to $f\left(a,b\right)=n$ admit a uniform three-parameter description -- recovering arithmetic-progression families via the Chinese remainder theorem when $n$ is squarefree. It shows that the density of pairs with $f\left(a,b\right)=1$ tends to ${\prod }_p\left(1-p^{-2}(p+1{)}^{-1}\right)\approx 0.88151$, and that its higher-order analogue $f_r$ has a limiting density $6/{\pi }^2$ for $r\ge 2$.

arXiv commentThere is an error regarding the computation of the limiting density