Geodesic
On the moments of elements of continued fractions for some rational numbers
Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 21, Tome 337 (2006), pp. 13-22 
E. P. Golubeva. On the moments of elements of continued fractions for some rational numbers. Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 21, Tome 337 (2006), pp. 13-22. http://geodesic.mathdoc.fr/item/ZNSL_2006_337_a1/
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     title = {On the moments of elements of continued fractions for some rational numbers},
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     url = {http://geodesic.mathdoc.fr/item/ZNSL_2006_337_a1/}
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

Let $p$ be a prime and let $1\le a\le p-1$. In the paper, an asymptotics for the sum over $a$ of the moments of order $\alpha$ ($0<\alpha<1$) of the sequence of elements of the expansion of $a/p$ into a continued fraction is obtained. As a corollary, an upper bound for the number of those $a$ whose expansions contain at least one element larger than $\log^\lambda p$ ($\lambda>1$) is derived. Note that in the case considered, the set of elements has no limiting distribution as $p\to\infty$, which is in contrast with the case of rational fractions $b/c$, where $(b,c)=1$ and $b^2+c^2\le R^2$ ($R\to\infty$). Bibliography: 6 titles.

[1] V. N. Popov, “Asimptotika summy summ elementov nepreryvnykh drobei chisel vida $a/p$”, Zap. nauchn. semin. LOMI, 91, 1979, 81–93 | MR | Zbl

[2] G. Myerson, “On semi-regular finite continued fractions”, Arch. Math., 48:5 (1987), 420–425 | DOI | MR | Zbl

[3] M. O. Avdeeva, V. A. Bykovskii, Reshenie zadachi Arnolda o statistikakh Gaussa–Kuzmina, Preprint/DVO, Khabarovskoe otdelenie IPM, No 8, 2002 | MR

[4] E. P. Golubeva, “O dlinakh periodov razlozheniya v nepreryvnuyu drob kvadratichnykh irratsionalnostei i chislakh klassov veschestvennykh kvadratichnykh polei, II”, Zap. nauchn. semin. LOMI, 168, 1988, 11–22 | MR | Zbl

[5] A. Ya. Khinchin, Tsepnye drobi, M., 1978 | MR | Zbl

[6] B. A. Venkov, Elementarnaya teoriya chisel, M.–L., 1937 | Zbl