On the Limiting Density of a gcd Map

Abstract

The function \[f(a,b)=\frac{\gcd(a+b,ab)}{\gcd(a,b)}\] is of interest in this paper. We then ask a natural question regarding how often $f(a,b)=1$ is. We yield the limiting density $\ddot{I}={\prod }_p\left(1-\frac{1}{p^2(p+1)}\right)\approx 0.88151$ which is an Euler product that unexpectedly matches the quadratic class number constant from the theory of real quadratic fields. We also consider its higher-order analogue $f_r$, where the problem collapses to coprimality and the density becomes $1/\hat{I}\P (2)=6/\ddot{I}{€}^2$.

There is an error regarding the computation of the limiting density