Ustinov, A. V. 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 \[ N(R)=N_1\log R+N_0+O(R^{-1+\varepsilon}), D(R)=D_1\log R +D_0+O(R^{-\frac{1}{3}+\varepsilon}). \] Reviewer: Florin Nicolae (Berlin) Cited in 2 Documents MSC: 11K50 Metric theory of continued fractions 11A55 Continued fractions 11L05 Gauss and Kloosterman sums; generalizations Keywords:continued fraction expansion; mean value; variance; asymptotic formulas PDFBibTeX XMLCite \textit{A. V. Ustinov}, Sb. Math. 198, No. 6, 887--907 (2007; Zbl 1197.11096); translation from Mat. Sb. 198, No. 6, 139--158 (2007) Full Text: DOI