# 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} }

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.

#### 8 Citations

On simultaneous diophantine approximations to $\zeta(2)$ and $\zeta(3)$

- Mathematics
- 2013

The authors present a hypergeometric construction of rational approximations to $\zeta(2)$ and $\zeta(3)$ which allows one to demonstrate simultaneously the irrationality of each of the zeta values,… Expand

Irrationality proofs for zeta values, moduli spaces and dinner parties

- Mathematics
- 2014

A simple geometric construction on the moduli spaces $\mathcal{M}_{0,n}$ of curves of genus $0$ with $n$ ordered marked points is described which gives a common framework for many irrationality… Expand

On the interplay between hypergeometric series, Fourier–Legendre expansions and Euler sums

- Mathematics, Physics
- 2018

In this work we continue the investigation, started in Campbell et al. (On the interplay between hypergeometric functions, complete elliptic integrals and Fourier–Legendre series expansions,… Expand

Do algebraic numbers follow Khinchin’s Law?∗

- 2021

This paper argues that the distribution of the coefficients of the regular continued fraction should be considered for each algebraic number of degree > 2 separately. For random numbers the… Expand

Hypergeometric rational approximations to ζ(4)

- Mathematics
- Proceedings of the Edinburgh Mathematical Society
- 2020

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… Expand

Hypergeometric heritage of W. N. Bailey

- Mathematics
- 2019

We review some of W.N. Bailey's work on hypergeometric functions that found solid applications in number theory. The text is complemented by Bailey's letters to Freeman Dyson from the 1940s.

Vectors of type II Hermite–Padé approximations and a new linear independence criterion

- Mathematics
- 2020

We propose a linear independence criterion, and outline an application of it. Down to its simplest case, it aims at solving this problem: given three real numbers, typically as special values of… Expand

#### References

SHOWING 1-10 OF 24 REFERENCES

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

Ramanujan-type formulae and irrationality measures of some multiples of $ {\pi}$

- Mathematics
- 2005

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 used… Expand

Well-poised hypergeometric service for diophantine problems of zeta values | NOVA. The University of Newcastle's Digital Repository

- Engineering, Mathematics
- 2003

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 de… Expand

Arithmetic of linear forms involving odd zeta values

- Mathematics
- 2002

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’s… Expand

On the irrationality exponent of the number ln 2

- Mathematics
- 2010

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 number… Expand

Irrationality of values of the Riemann zeta function

- Mathematics
- 2002

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 prove… Expand

Generalized hypergeometric series

- Mathematics
- 1935

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

- Mathematics
- 2013

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-duality… Expand

A few remarks on ζ(3)

- Mathematics
- 1996

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 of… Expand

Арифметические гипергеометрические ряды@@@Arithmetic hypergeometric series

- Mathematics
- 2011

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. Originally… Expand