The group structure for ζ(3)

@article{Rhin2001TheGS,
  title={The group structure for $\zeta$(3)},
  author={Georges Rhin and Carlo Viola},
  journal={Acta Arithmetica},
  year={2001},
  volume={97},
  pages={269-293}
}
1. Introduction. In his proof of the irrationality of ζ(3), Apéry [1] gave sequences of rational approximations to ζ(2) = π 2 /6 and to ζ(3) yielding the irrationality measures µ(ζ(2)) < 11.85078. .. and µ(ζ(3)) < 13.41782. .. Several improvements on such irrationality measures were subsequently given, and we refer to the introductions of the papers [3] and [4] for an account of these results. As usual, we denote here by µ(α) the least irrationality measure of an irrational number α, i.e., the… Expand
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References

SHOWING 1-6 OF 6 REFERENCES
A Note on the Irrationality of ζ(2) and ζ(3)
At the “Journees Arithmetiques” held at Marseille-Luminy in June 1978, R. Apery confronted his audience with a miraculous proof for the irrationality of ζ(3) = l-3+ 2-3+ 3-3 + .... The proof wasExpand
Irrationalité de ζ(2) et ζ(3)
  • Astérisque 61
  • 1979
Ile du Saulcy 57045 Metz Cedex 01, France E-mail: rhin@poncelet.univ-metz.fr Dipartimento di Matematica Università di Pisa Via Buonarroti 2 56127 Pisa, Italy E-mail: viola@dm.unipi.it Received on 23
  • Ile du Saulcy 57045 Metz Cedex 01, France E-mail: rhin@poncelet.univ-metz.fr Dipartimento di Matematica Università di Pisa Via Buonarroti 2 56127 Pisa, Italy E-mail: viola@dm.unipi.it Received on 23
  • 1999
Astérisque
  • Astérisque
  • 1979