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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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