A new irrationality measure for \(\zeta(3)\). (English) Zbl 0955.11023

The author proves that \(\zeta(3)\) has irrationality measure \(\mu= 7.377956\dots\), thus improving earlier results [R. Dvornicich and C. Viola, Colloq. Math. Soc. János Bolyai 51, 637-657 (1987; Zbl 0755.11019); M. Hata, J. Reine Angew. Math. 407, 99-125 (1990; Zbl 0692.10034)]. The proof rests on Beuker’s integral [F. Beukers, Bull. Lond. Math. Soc. 11, 268-272 (1979; Zbl 0241.10023)] and uses Legendre-type polynomials in the form \[ L_{n,m}(x)= \tfrac{1}{n!} (x^{n+m} (1-x)^n)^{(n)}. \]
Reviewer: D.Duverney (Lille)


11J82 Measures of irrationality and of transcendence
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
11M06 \(\zeta (s)\) and \(L(s, \chi)\)
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