Asymptotic behaviour of the first and second moments for the number of steps in the Euclidean algorithm

We prove asymptotic formulae with two significant terms for the expectation and variance of the random variable $s(c/d)$ when the variables $c$ and $d$ range over the set $1\leq c\leq d\leq R$ and $R\to\infty$, where $s(c,d)=s(c/d)$ is the number of steps in the Euclidean algorithm applied to the numbers $c$ and $d$.