Sorokin, V. N. Hermite-Padé approximations of Nikishin systems and the irrationality of \(\zeta(3)\). (English. Russian original) Zbl 0827.11042 Russ. Math. Surv. 49, No. 2, 176-177 (1994); translation from Usp. Mat. Nauk 49, No. 2(296), 167-168 (1994). The author constructs a new type of Hermite-Padé approximation of Nikishin systems [E. M. Nikishin, Sov. Math. 30, 43-52 (1986); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1986, No. 2, 33-41 (1986; Zbl 0631.30036)] and announces several assertions on the properties of such approximations, omitting proofs. As an interesting application, he obtains that \(\lim_{n\to\infty} r_n^{1/n}= e^3 (\sqrt{2}- 1)^4= 0.59\dots\), where \(r_n= 2q_n \zeta(3)- p_n\neq 0\) \((p_n, q_n\in \mathbb{Z})\) is the Apéry approximation of \(\zeta (3)\) [R. Apéry, Astérisque 61, 11-13 (1979; Zbl 0401.10049)]. Reviewer: Zhu Yaochen (Beijing) Cited in 10 Documents MSC: 11J82 Measures of irrationality and of transcendence 41A21 Padé approximation Keywords:irrationality; Hermite-Padé approximations; Nikishin systems; Apéry approximation Citations:Zbl 0631.30036; Zbl 0401.10049 PDFBibTeX XMLCite \textit{V. N. Sorokin}, Russ. Math. Surv. 49, No. 2, 1 (1994; Zbl 0827.11042); translation from Usp. Mat. Nauk 49, No. 2(296), 167--168 (1994) Full Text: DOI