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The group structure for \(\zeta(3)\). (English) Zbl 1004.11042
R. Apéry [Astérisque 61, 11-13 (1979; Zbl 0401.10049)] proved the irrationality of \(\zeta(3)\) and gave the irrationality measure \(\mu(\zeta(3))< 13.41782\dots\;\). The authors [Acta Arith. 77, 23-56 (1996; Zbl 0864.11037)] obtained the record irrationality measure for \(\zeta(2)\), viz. \(\mu(\zeta(2))< 5.441243\dots\) by an arithmetical study of a family of double integrals lying in \(\mathbb{Q}+ \mathbb{Z}\zeta(2)\). In the present paper, the authors succeed in adapting their method to a family of triple integrals lying in \(\mathbb{Q}+ \mathbb{Z}\zeta(3)\) and obtain \(\mu(\zeta(3))< 5.513891\dots\;\).
The triple integrals are given by \[ \int_0^1 \int_0^1 \int_0^1 \frac {x^h(1-x)^ly^k (1-y)^sz^j(1-z)^q} {(1-(1-xy)z)^{q+h-r}} \frac {dx dy dz} {1-(1-xy)z}. \] The birational transformation \[ X= (1-y)z, \qquad Y= \frac{(1-x)(1-z)} {1-(1-xy)z}, \qquad Z= \frac{y}{1-(1-y)z} \] of period 8 produces a cyclic permutation of the 8 parameters in the triple integral and provides the basis of the algebraic structure at the heart of the proof.

MSC:
11J82 Measures of irrationality and of transcendence
11M06 \(\zeta (s)\) and \(L(s, \chi)\)
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