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

A. V. Ustinov

Sbornik. Mathematics, Tome 198 (2007) no. 6, pp. 887-907

Résumé

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 $$ N(R)=N_1\log R+N_0+O(R^{-1+\varepsilon}), \quad D(R)=D_1\log R+D_0+O(R^{-1/3+\varepsilon}). $$ Bibliography: 13 titles.