Two hypergeometric tales and a new irrationality measure of $$\zeta (2)$$ζ(2)

@article{Zudilin2013TwoHT,
  title={Two hypergeometric tales and a new irrationality measure of \$\$\zeta (2)\$\$$\zeta$(2)},
  author={W. Zudilin},
  journal={Annales math{\'e}matiques du Qu{\'e}bec},
  year={2013},
  volume={38},
  pages={101-117}
}
  • W. Zudilin
  • Published 2013
  • Mathematics
  • Annales mathématiques du Québec
We prove the new upper bound $$5.095412$$5.095412 for the irrationality exponent of $$\zeta (2)=\pi ^2/6$$ζ(2)=π2/6; the earlier record bound $$5.441243$$5.441243 was established in 1996 by G. Rhin and C. Viola.RésuméNous obtenons une nouvelle borne pour l’exposant d’irrationnalité de $$\zeta (2)=\pi ^2/6$$ζ(2)=π2/6, à savoir $$5.095412$$5.095412, cette dernière améliorant le record $$5.441243$$5.441243 établi par G. Rhin et C. Viola. 
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References

SHOWING 1-10 OF 24 REFERENCES
The group structure for ζ(3)
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. ..Expand
Ramanujan-type formulae and irrationality measures of some multiples of $ {\pi}$
An explicit construction of simultaneous Pade approximations for gener- alized hypergeometric series and formulae for the quantities π √ d , d ∈{ 1, 2, 3, 10005}, in terms of these series are usedExpand
Well-poised hypergeometric service for diophantine problems of zeta values | NOVA. The University of Newcastle's Digital Repository
On montre comment les concepts classiques de series et integrales hypergeometriques bien equilibrees devient crucial dans l'etude des proprietes arithmetiques des valeurs de la fonction zeta deExpand
Arithmetic of linear forms involving odd zeta values
The story exposed in this paper starts in 1978, when R. Apery [Ap] gave a surprising sequence of exercises demonstrating the irrationality of ζ(2) and ζ(3). (For a nice explanation of Apery’sExpand
On the irrationality exponent of the number ln 2
We propose another method of deriving the Marcovecchio estimate for the irrationality measure of the number ln 2 following, for the most part, the method of proof of the irrationality of the numberExpand
Irrationality of values of the Riemann zeta function
The paper deals with a generalization of Rivoal's construction, which enables one to construct linear approximating forms in 1 and the values of the zeta function ζ(s) only at odd points. We proveExpand
Generalized hypergeometric series
This also gives in the paper T. H. Koornwinder, Orthogonal polynomials with weight function (1− x)α(1 + x)β + Mδ(x + 1) + Nδ(x− 1), Canad. Math. Bull. 27 (1984), 205–214 the identitity (2.5) with N =Expand
WZ-proofs of "divergent" Ramanujan-type series
We prove some “divergent” Ramanujan-type series for \(1/\pi\) and \(1{/\pi }^{2}\) applying a Barnes-integrals strategy of the WZ-method. In addition, in the last section, we apply the WZ-dualityExpand
A few remarks on ζ(3)
A new proof of the irrationality of the number ζ(3) is proposed. A new decomposition of this number into a continued fraction is found. Recurrence relations are proved for some sequences ofExpand
Арифметические гипергеометрические ряды@@@Arithmetic hypergeometric series
The main goal of our survey is to give common characteristics of auxiliary hypergeometric functions (and their generalisations), functions which occur in number-theoretical problems. OriginallyExpand
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