On the statistical properties of finite continued fractions
type: Item
instance of: scholarly article
title: On the statistical properties of finite continued fractions
zbMATH Open document ID: 1072.11060
author: Alexey Ustinov
published in: Journal of Mathematical Sciences (New York)
publication date: 2005-07-14T00:00:00Z
full work available at URL: https://eudml.org/doc/227267
review text: 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.
reviewed by: Mykhaylo Pahirya
Mathematics Subject Classification ID: 11K50
Mathematics Subject Classification ID: 11A55
zbMATH DE Number: 2187246
zbMATH Keywords: continued fractions
zbMATH Keywords: metric theory
MaRDI profile type: Publication
OpenAlex ID: W2072601494
cites work: Fermat dynamics, matrix arithmetics, finite circles, and finite Lobachevsky planes
cites work: Q4398780
cites work: The number of steps in the Euclidean algorithm
cites work: Q5613266
cites work: Evaluation of Porter's constant
cites work: Q3916824
cites work: On a theorem of Heilbronn
DOI: 10.1007/S10958-006-0268-6
description: scientific article; zbMATH DE number 2187246