Abstract:
We study the random variable
$N(\alpha,R)=\#\{j\geqslant1:Q_j(\alpha)\leqslant R\}$,
where $\alpha\in[0;1)$ and $P_j(\alpha)/Q_j(\alpha)$ is the
$j$th convergent of the continued fraction expansion of the number
$\alpha=[0;t_1,t_2,\dots]$. For the mean value
$$
N(R)=\int_0^1N(\alpha,R)\,d\alpha
$$
and variance
$$
D(R)=\int_0^1\bigl(N(\alpha,R)-N(R)\bigr)^2\,d\alpha
$$
of the random
variable $N(\alpha,R)$, we prove the asymptotic formulae with two
significant terms
$$
N(R)=N_1\log R+N_0+O(R^{-1+\varepsilon}), \quad
D(R)=D_1\log R+D_0+O(R^{-1/3+\varepsilon}).
$$
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