Geodesic
Asymptotic behaviour of the first moment of the number of steps in the by-excess and by-deficiency Euclidean algorithms
Sbornik. Mathematics, Tome 203 (2012) no. 2, pp. 288-305

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The first moments for the number of steps in different Euclidean algorithms are considered. For these moments asymptotic formulae with new remainder terms are obtained using refined estimates for sums of fractional parts and some ideas in Selberg's elementary proof of the prime number theorem. Bibliography: 12 titles.
Keywords: Euclidean algorithms, continued fractions, fractional parts, prime number theorem. 
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     author = {D. Frolenkov},
     title = {Asymptotic behaviour of the first moment of the number of steps in the by-excess and by-deficiency {Euclidean} algorithms},
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D. Frolenkov. Asymptotic behaviour of the first moment of the number of steps in the by-excess and by-deficiency Euclidean algorithms. Sbornik. Mathematics, Tome 203 (2012) no. 2, pp. 288-305. http://geodesic.mathdoc.fr/item/SM_2012_203_2_a6/