Elementary proof of Yu. V. Nesterenko expansion of the number zeta(3) in continued fraction
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 \(\theta = 1, 2, 3\).
scientific article; zbMATH DE number 5690836
scientific article