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.


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