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Calculation of the variance in a problem in the theory of continued fractions. (English) Zbl 1197.11096

Sb. Math. 198, No. 6, 887-907 (2007); translation from Mat. Sb. 198, No. 6, 139-158 (2007).
The author studies the random variable \[ N(\alpha,R)=\#\{j\geq 1:Q_j(\alpha)\leq R\}, \] where \(\alpha\in[0;1)\) and \(\frac{P_j(\alpha)}{Q_j(\alpha)} \) is the \(j\)-th convergent of the continued fraction expansion of the number \(\alpha=[0;t_1,t_2,\ldots]\). For the mean value \[ N(R)=\int_0^1 N(\alpha,R)d\alpha \] and variance \[ D(R)=\int_0^1(N(\alpha,R)-N(R))^2 d\alpha \] he proves the asymptotic formulae \[ N(R)=N_1\log R+N_0+O(R^{-1+\varepsilon}), D(R)=D_1\log R +D_0+O(R^{-\frac{1}{3}+\varepsilon}). \]

MSC:

11K50 Metric theory of continued fractions
11A55 Continued fractions
11L05 Gauss and Kloosterman sums; generalizations
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