On the statistical properties of finite continued fractions
The article is devoted to the statistical properties of continued fractions for numbers \(a/b\). The numbers \(a\) and \(b\) are contained in the sector \((a,b \geq 1, \, a^2+b^2 \leq R^2)\). The main result is an asymptotic formula for \[ N_x(R)=\sum_{\substack{ a^2+b^2 \leq R^2 \\ a, b \in \mathbb{N}}} s_x(a/b)=\frac3{\pi}R^2[\log(1+x)R^2\log R+C(x)]+O(R^{17/9}\log^2R), \] where \(s_x(a/b)=|\{j\in \{1,2,\ldots,s\} : [0;t_j,\ldots,t_s]\leq x \}|\) is the Gaussian statistic for the fraction \(\frac ab=[t_0,t_1,\ldots,t_s]\) and \(C(x)\) a complicated function. The bibliography contains 12 items.
scientific article; zbMATH DE number 2187246