# Multiple zeta values, Pad\'e approximation and Vasilyev's conjecture

@article{Fischler2013MultipleZV, title={Multiple zeta values, Pad\'e approximation and Vasilyev's conjecture}, author={St{\'e}phane Fischler and Tanguy Rivoal}, journal={arXiv: Number Theory}, year={2013} }

Sorokin gave in 1996 a new proof that pi is transcendental. It is based on a simultaneous Pad\'e approximation problem involving certain multiple polylogarithms, which evaluated at the point 1 are multiple zeta values equal to powers of pi. In this paper we construct a Pad\'e approximation problem of the same flavour, and prove that it has a unique solution up to proportionality. At the point 1, this provides a rational linear combination of 1 and multiple zeta values in an extended sense that…

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## References

SHOWING 1-10 OF 45 REFERENCES

Multiple series connected to Hoffman's conjecture on multiple zeta values

- Mathematics
- 2006

Abstract Recent results of Zlobin and Cresson–Fischler–Rivoal allow one to decompose any suitable p-uple series of hypergeometric type into a linear combination (over the rationals) of multiple zeta…

Irrationalit\'e de valeurs de z\^eta (d'apr\`es Ap\'ery, Rivoal, ...)

- Mathematics
- 2003

This survey text deals with irrationality, and linear independence over the rationals, of values at positive odd integers of Riemann zeta function. The first section gives all known proofs (and…

An identity of Andrews, multiple integrals, and very-well-poised hypergeometric series

- Mathematics
- 2003

We give a new proof of a theorem of Zudilin that equates a very-well-poised hypergeometric series and a particular multiple integral. This integral generalizes integrals of Vasilenko and Vasilyev…

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…

Multiple integrals and linear forms in zeta-values

- Mathematics
- 2007

We deflne n-dimensional Beukers-type integrals over the unit hypercube. Using an n-dimensional birational transformation we show that such integrals are equal to suita- ble n-dimensional Sorokin-type…

TRANSCENDENTAL NUMBERS

- 2012

The Greeks tried unsuccessfully to square the circle with a compass and straightedge. In the 19th century, Lindemann showed that this is impossible by demonstrating that π is not a root of any…

On small linear forms for the values of the Riemann zeta-function at odd points

- Mathematics
- 2014

If dn = lcm (1, 2, . . . , n) then the irrationality of ζ(3) was obtained by showing a) For any nonnegative integer n we have dnJn = An −Bnζ(3) where An, Bn ∈ Z. b) 0 0 and α are constants with 0 2…

Phénomènes de symétrie dans des formes linéaires en polyzêtas

- Physics, Mathematics
- 2006

Abstract We give two generalizations, in arbitrary depth, of the symmetry phenomenon used by Ball-Rivoal to prove that infinitely many values of Riemann ζ function at odd integers are irrational.…

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. ..…

A transcendence measure for π2

- Mathematics
- 1996

A new proof of the fact that π2 is transcendental is proposed. A modification of Hermite's method for an expressly constructed Nikishin system is used. The Beukers integral, which was previously used…