On the Limiting Density of a gcd Map

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 $\rho ={\prod }_p(1-\frac{1}{p^2(p+1)})\approx 0.88151$ 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 $1\mathrm{/}\zeta (2)=6\mathrm{/}{\pi }^2$.