Geodesic
The average length of finite continued fractions with fixed denominator
Sbornik. Mathematics, Tome 208 (2017) no. 5, pp. 644-683

Voir la notice de l'article provenant de la source Math-Net.Ru

In 1968 Heilbronn proved an asymptotic formula for the mean value of the lengths of continued fraction expansions of rational numbers with identical denominators. A new method is proposed for solving Heilbronn's problem and its generalizations. New estimates for the remainders, which improve the earlier results due to Porter (1975) and Ustinov (2005), are obtained. Bibliography: 28 titles.
Keywords: continued fraction, additive divisor problem
Mots-clés : convolution formula. 
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V. A. Bykovskii; D. A. Frolenkov. The average length of finite continued fractions with fixed denominator. Sbornik. Mathematics, Tome 208 (2017) no. 5, pp. 644-683. http://geodesic.mathdoc.fr/item/SM_2017_208_5_a2/