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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)

##### MSC:
 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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