On the moments of elements of continued fractions for some rational numbers

E. P. Golubeva

Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 21, Tome 337 (2006), pp. 13-22

Résumé

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 $\alpha$ ($0\alpha1$) 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.