The convergence worsens a lot for any $\gamma$ if summing over all integer points below the hyperbola (without the factor $\zeta(2)$). It looks surprisingly good only when summing over coprimes for $\gamma=\frac{1}{2}$. It may have something to do with the conjecture I have posted there (which may be more suitable for MO...)?

As a consequence of your argument, we can also notice that any weight of the form $2(1-\gamma)\zeta(2)(ab)^{-\gamma}n^{\gamma-1}$ should work just as fine. However, I observe that the convergence is faster for $\gamma=\frac{1}{2}$. Any heuristic for that?

Thank you for the additional hints. I still find the details of the last paragraph slightly confusing, like the relation between $\Omega_{s,r,c}$ or the integral $\int_{x,y>0}\frac{dxdy}{\sqrt{xy}}$ which in fact does not converge. But the argument about the uniform distribution of coprime and the last integral teached me a lot!

Nice explanation! I am trying to understand to point of splitting the sum as you propose. Can't we simply skip this step? The final integral explains well the role of the condition $ab<n$. Also, this holds assuming that the coprime points are indeed uniformly distributed in the plane. However, the reference I have found (Hardy & Wright, section 18.5) only states the average density in a square $\max(a,b)\leq n$. A more precise reference would complete nicely your argument.

This is precisely my interrogation. The Farey enumeration (with inverses) of order $n^2$ with weights $\frac{1}{n\sqrt{ab}}$ converges to another measure $e^{-\frac{1}{2}|x|}\textrm{d}x$.

"An Introduction to the Theory of Numbers" from Hardy & Wright (section 18.5) speaks about the density $1/\zeta(2)$.

It is probably related to this uniformity in a way or another, but the detail in not clear to me. $\zeta(2)$ can be factored out of a more brute expression $\sum_{k,l=0}^\infty \frac{F(\log \frac{k}{l})}{kl}$ (which does not converge). The fact that we have to trim the rationals with $ab>n$ make the enumeration of rationals different from (but included in) the union of the Farey sequence and the set of inverses, up to order $n^2$.

Yes, I have introduced $F$ as a test function here, to formulate the question about the underlying empirical distribution, which is seemingly asymptotically uniform and equal to 1. I wonder how close this measure is to the Lebesgue measure.

I mean Jordan's totient function, but I'm not sure this helps to compute the remaining sums over factors of $c$ and $d$.