Calculation of the variance in a problem in the theory of continued fractions

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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