While reading Steven Finch's book Mathematical constants (I believe), I once came across and wrote down the following theorem: For almost all real numbers $x$, if $\frac{P_n}{Q_n}$ is the $n^{\text{th}}$ partial convergent of the continued fraction of $x$, then:
$$\lim_{n\rightarrow\infty}Q_n^{\;\frac{1}{n}}=e^{\frac{\pi^2}{12\ln{2}}}\tag{*}$$
This seems like a fascinating result; presumably the $\ln{2}$ in the exponent comes from the same place as the base $2$ logarithm in the expressions for the Gauss-Kuzmin distribution and Khinchin's constant, although I am not too familiar with the asymptotics of continued fractions. The factor of $\frac{\pi^2}{12}$ seems more unusual; does it perhaps come from $\zeta(2)$, e.g. something to do with the fact that $\zeta(2)$ is related to the probability of two randomly chosen integers being coprime, or perhaps there is just some arcane way of transforming the LHS of $(*)$ to get $\zeta(2)$ involved? I am therefore curious for a general idea of the methods that would be used to prove this result, or even a sketch of such a proof (I no longer have access to the book, and I haven't been able to find a reference for this formula online by searching all the relevant words I can think of, so I'm not sure where to look).
Thus my question is: By what methods can $(*)$ be proved, and is there a simple elegant way of proving it?