In this article (a pdf is easily findable online) Zagier gives an approximation of the number of Markoff number below a certain bound. What I struggle with is to show that if \begin{equation} E_{\alpha,\beta}(x) = \frac{3x^2}{\pi^2\alpha\beta} + O\left( \frac{x}{\alpha} \right) + O\left( \frac{x}{\beta} \log \frac{x}{\beta} \right) \end{equation} And \begin{equation} E_{f(p),f(q)}(f(x)) \leq M_m(x) \leq E_{f(p)-\frac{c}{r^2},f(q)- \frac{c}{r^2} }(f(x)) \end{equation} Then \begin{equation} M_m(x) = \frac{3f(x)^2}{\pi^2(f(p)+O(1/r^2)) (f(q)+O(1/r^2))}+ O\left( \frac{f(x)}{f(p)} \right) + O\left( \frac{f(x)}{f(q)} \log \frac{f(x)}{f(q)} \right) \end{equation}
Where $m=(p,q,r)$ is a Markoff triple, $f(x)=arccosh(3x/2)$ and c is a constant such that \begin{equation} f(r)<f(p)+f(q)<f(r)+\frac{c}{r^2} \end{equation}
More details can be found in the article but I believe that this is enough. Thanks a lot if you take time to help me.