Elementary proof of Yu. V. Nesterenko expansion of the number zeta(3) in continued fraction
type: Item
instance of: scholarly article
title: Elementary proof of Yu. V. Nesterenko expansion of the number zeta(3) in continued fraction
zbMATH Open document ID: 1223.11086
DOI: 10.1155/2010/143521
published in: Advances in Difference Equations
publication date: 2010-04-08T00:00:00Z
full work available at URL: https://eudml.org/doc/225577
review text: In 1978 R. Apéry proved his famous result for the irrationality of \(\zeta(3)\) by using the presentation of \(\zeta(3)\) in continued fraction \[ \zeta(3) = \frac{6|}{|5}-\frac{1|}{|117}-\ldots-\frac{n^{6}|}{|34n^{3}+51n^{2}+27n+5}-\ldots, \] showing, that \[ \left|\zeta(3)-\frac{p}{q}\right|<\frac{1}{q^{\mu}} \] for some \(\mu>13\), has a finite number of solutions in natural \(p\) and \(q\). In 1996 \textit{Yu. V. Nesterenko} proved [Math. Notes 59, No. 6, 625--636 (1996); translation from Mat. Zametki 59, No. 6, 865--880 (1996; Zbl 0888.11028)] that \[ 2\zeta(3) =2 +\frac{1|}{|2}+\frac{2|}{|4}+\frac{1|}{|3}+\frac{4|}{|2}+\frac{2|}{|4}+\frac{6|}{|0}+\frac{4|}{|5}+\frac{9|}{|4}+\frac{6|}{|6}+\frac{12|}{|8}+\frac{9|}{|7}+\frac{16|}{|6}+\ldots+\frac{a_{n}}{b_{n}}\ldots, \] where \(a_{1}=1, a_{4k+1}=k(k+1), a_{4k+2}=(k+1)(k+2), a_{4k+3}=(k+1)^{2}, a_{4k+4}=(k+2)^{2}\) for \(k\geq 0, 4k+1\geq 2\) and \(b_{4k+1}=2k+2, b_{4k+2}=2k+4, b_{4k+3}=2k+3, b_{4k+4}=2k+2\) for \(k\geq 0\). He proved the result of Apéry by using integrals and hypergeometric functions, in particular G-functions of Meijer, with many recurrent relations. By using difference equations, the author of the present paper proposes a more elementary proof, but with many details, of the expression of \(\zeta(3)\) found by Nesterenko and for the approximation to \(\zeta(3)\) he proves that there exist constants \(c_{3}(\varepsilon)>0, c_{4}(\varepsilon)>0, \varepsilon\) for which \[ \frac {c_{3}(\varepsilon)}{(1+\sqrt{2})^{8k(k+\varepsilon}}<\left|\zeta(3)-\frac{P_{4k-2+\theta}}{Q_{4k-2}}\right|<\frac{{c}_{4}(\varepsilon)}{(1+\sqrt{2})^{8k(1-\varepsilon}} \] for some constants \(c_{3}(\varepsilon) > 0,c_{4}(\epsilon) > 0, \varepsilon > 0\) and \(\theta = 1, 2, 3\).
reviewed by: Tonko Tonkov
Mathematics Subject Classification ID: 11J70
Mathematics Subject Classification ID: 11Y60
Mathematics Subject Classification ID: 11A55
zbMATH DE Number: 5690836
zbMATH Keywords: difference equations
zbMATH Keywords: continued fractions
zbMATH Keywords: irrationality
Wikidata QID: Q59250074
author: Leonid A. Gutnik
MaRDI profile type: Publication
OpenAlex ID: W2063753505
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description: scientific article; zbMATH DE number 5690836