# Hypergeometric rational approximations to ζ(4)

@article{Marcovecchio2020HypergeometricRA,
title={Hypergeometric rational approximations to $\zeta$(4)},
journal={Proceedings of the Edinburgh Mathematical Society},
year={2020},
volume={63},
pages={374 - 397}
}
• Published 2020
• Mathematics
• Proceedings of the Edinburgh Mathematical Society
Abstract We give a new hypergeometric construction of rational approximations to ζ(4), which absorbs the earlier one from 2003 based on Bailey's 9F8 hypergeometric integrals. With the novel ingredients we are able to gain better control of the arithmetic and produce a record irrationality measure for ζ(4).
2 Citations
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• 2021
A classical integrable N -body system representing a relativistic generalization of the Calogero and Sutherland models has been suggested by Ruijsenaars and Schneider [20]. Its quantizationExpand
A case study for $\zeta(4)$
• Mathematics, Computer Science
• 2020
Using symbolic summation tools in the setting of difference rings, we prove a two-parametric identity that relates rational approximations to $\zeta(4)$.

#### References

SHOWING 1-10 OF 35 REFERENCES
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
Hypergeometry inspired by irrationality questions
• Mathematics
• 2018
We report new hypergeometric constructions of rational approximations to Catalan's constant, $\log2$, and $\pi^2$, their connection with already known ones, and underlying `permutation group'Expand
WELL-POISED HYPERGEOMETRIC TRANSFORMATIONS OF EULER-TYPE MULTIPLE INTEGRALS
Several new multiple-integral representations are proved for well-poised hypergeometric series and integrals. The results yield, in particular, transformations of the multiple integrals that cannotExpand
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
The group structure for ζ(3)
• Mathematics
• 2001
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
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
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
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
Two hypergeometric tales and a new irrationality measure of $$\zeta (2)$$ζ(2)
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. RhinExpand
La fonction zêta de Riemann prend une infinité de valeurs irrationnelles aux entiers impairs
Resume Nous montrons que la dimension de l'espace vectoriel engendre sur les rationnels par 1 et les n premieres valeurs de la fonction zeta de Riemann aux entiers impairs croit au moins comme unExpand