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

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.