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 $1\le c\le d\le R\infty$. 
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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/

[1] H. Minkowski, “Zur Theorie der Kettenbruche”, Annales de l'Ecole Normale Superieure, 13:3 (1894), 41–60 | MR

[2] B. Vallée, “Dynamical analysis of a class of Euclidean Algorithms”, Theoret. Comput. Sci., 297 (2003), 447–486 | DOI | MR | Zbl

[3] Y. Hartono, Ergotic properties of continued fraction algorithms, Thesis (Dr.)–Technische Universiteit Delft (The Netherlands), 2003, 119 pp. | MR

[4] J. W. Porter, “On a theorem of Heilbronn”, Mathematika, 22:1 (1975), 20–28 | DOI | MR | Zbl

[5] D. E. Knuth, “Evaluation of Porter's Constant”, Comp. and Maths. with Appls., 2 (1976), 137–139 | DOI | MR | Zbl

[6] G. H. Norton, “On the asymptotic analysis of the Euclidean algorithm”, J. Symbolic Comput., 10:1 (1990), 53–58 | DOI | MR | Zbl

[7] A. V. Ustinov, “Asimptoticheskoe povedenie pervogo i vtorogo momenta dlya chisla shagov v algoritme Evklida”, Izvestiya RAN, 72:5 (2008), 189–224 | MR | Zbl

[8] V. Baladi, B. Vallée, “Euclidean algorithms are Gaussian”, J. Number Theory, 110 (2005), 331–386 | DOI | MR | Zbl

[9] A. V. Ustinov, “O srednem chisle shagov v algoritme Evklida s vyborom minimalnogo po modulyu ostatka”, Matematicheskie zametki, 85:1 (2009), 153–156 | DOI | MR | Zbl

[10] A. V. Ustinov, “O srednem chisle shagov v algoritme Evklida s nechetnymi nepolnymi chastnymi”, Matematicheskie zametki, 88:4 (2010), 594–604 | DOI | MR | Zbl

[11] O.A. Gorkusha, “O konechnykh tsepnykh drobyakh spetsialnogo vida”, Chebyshevskii sbornik, 9:1(25) (2008), 80–108 | MR

[12] A. A. Karatsuba, Osnovy analiticheskoi teorii chisel, Nauka, M., 1983, 500 pp. | MR

[13] A. Ya. Khinchin, Izbrannye trudy po teorii chisel, MTsNMO, M., 2006, 260 pp. | MR

[14] D. E. Knuth, A. C. Yao, “Analysis of the Subtractive Algorithm for Greatest Common Divisors”, Proc. Nat. Acad. Sci. USA, 72:12 (1975), 4720–4722 | DOI | MR | Zbl