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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)].

MSC:

11J82 Measures of irrationality and of transcendence
41A21 Padé approximation
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