Geodesic
Asymptotic behaviour of the first and second moments for the number of steps in the Euclidean algorithm
Izvestiya. Mathematics , Tome 72 (2008) no. 5, pp. 1023-1059

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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 $d$. 
@article{IM2_2008_72_5_a5,
     author = {A. V. Ustinov},
     title = {Asymptotic behaviour of the first and second moments for the number of steps in the {Euclidean} algorithm},
     journal = {Izvestiya. Mathematics },
     pages = {1023--1059},
     publisher = {mathdoc},
     volume = {72},
     number = {5},
     year = {2008},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2008_72_5_a5/}
}
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A. V. Ustinov. Asymptotic behaviour of the first and second moments for the number of steps in the Euclidean algorithm. Izvestiya. Mathematics , Tome 72 (2008) no. 5, pp. 1023-1059. http://geodesic.mathdoc.fr/item/IM2_2008_72_5_a5/