Geodesic
The average length of Minkowski's diagonal continued fractions
Dalʹnevostočnyj matematičeskij žurnal, Tome 11 (2011) no. 1, pp. 10-27

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We prove asymptotic formulae with two significant terms for the expectation of the random variable $s(c/d;1)$ — length of Minkowski's diagonal continued fraction when the variables $c$ and $d$ range over the set $1\le c\le d\le R\infty$. 
@article{DVMG_2011_11_1_a1,
     author = {O. A. Gorkusha},
     title = {The average length of {Minkowski's} diagonal continued fractions},
     journal = {Dalʹnevosto\v{c}nyj matemati\v{c}eskij \v{z}urnal},
     pages = {10--27},
     publisher = {mathdoc},
     volume = {11},
     number = {1},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DVMG_2011_11_1_a1/}
}
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O. A. Gorkusha. The average length of Minkowski's diagonal continued fractions. Dalʹnevostočnyj matematičeskij žurnal, Tome 11 (2011) no. 1, pp. 10-27. http://geodesic.mathdoc.fr/item/DVMG_2011_11_1_a1/