# Arithmetic of linear forms involving odd zeta values

@article{Zudilin2002ArithmeticOL, title={Arithmetic of linear forms involving odd zeta values}, author={W. Zudilin}, journal={Journal de Theorie des Nombres de Bordeaux}, year={2002}, volume={16}, pages={251-291} }

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 discovery we refer to the review [Po].) Although the irrationality of the even zeta values ζ(2), ζ(4), . . . for that moment was a classical result (due to L. Euler and F. Lindemann), Apery’s proof allows one to obtain a quantitative version of his result, that is, to evaluate irrationality exponents:

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#### References

SHOWING 1-10 OF 105 REFERENCES

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

A Note on the Irrationality of ζ(2) and ζ(3)

- Mathematics
- 1979

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 was… Expand

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

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

Rational approximations to the dilogarithm

- Mathematics
- 1993

The irrationality proof of the values of the dilogarithmic function L 2 (z) at rational points z = 1/k for every integer k ∈ (−∞, −5] ∪ [7, ∞) is given. To show this we develop the method of… Expand

A note on Beukers' integral

- Mathematics
- 1995

The aim of this note is to give a sharp lower bound for rational approximations to ζ(2) = π 2 /6 by using a specific Beukers' integral. Indeed, we will show that π 2 has an irrationality measure less… Expand

A Proof that Euler Missed...

- Mathematics
- 2000

The board of programme changes informed us that R. Apery (Caen) would speak Thursday, 14.00 “Sur l’irrationalite de ζ(3).” Though there had been earlier rumours of his claiming a proof, scepticism… Expand

Cancellation of factorials

- Mathematics
- 2000

We study the arithmetic property which allows to sharpen number-theoretic estimates. Previous results on this property are, as a rule, quantitive. The application of our general qualitive theorems to… Expand

ON IRRATIONALITY OF THE VALUES OF THE FUNCTIONS $ F(x,s)$

- Mathematics
- 1980

This paper gives a proof of a theorem on the linear independence over of values of functions , where is a rational number whose numerator and denominator satisfy a certain relation.Bibliography: 1… Expand

On irrationality measures of the values of Gauss hypergeometric function

- Mathematics
- 1993

The paper gives irrationality measures for the values of some Gauss hypergeometric functions both in the archimedean andp-adic case. Further, an improvement of general results is obtained in the case… Expand