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## Christian Krattenthaler and
Tanguy Rivoal

# Hypergéométrie et fonction zêta de Riemann

### (73 pages)

**English Abstract.**
We prove the second author's "denominator
conjecture" [40] concerning the common denominators of
coefficients of certain linear forms in zeta values. These forms were recently
constructed to obtain lower bounds for the dimension of the vector space over
*Q* spanned by
1,*\zeta*(*m*),*\zeta*(*m*+2),...,*\zeta*(*m*+2*h*),
where *m* and *h*
are integers such that *m*>=2 and *h*>=0. In particular, we
immediately get the following results as corollaries: at least one
of the eight numbers
*\zeta*(5),*\zeta*(7),...,*\zeta*(19) is irrational, and
there exists an odd integer *j* between 5 and 165 such that 1,
*\zeta*(3) and *\zeta*(*j*) are
linearly independent over *Q*. This strengthens
some recent results in [41] and [8], respectively. We also
prove a related conjecture, due to Vasilyev [49],
and as well a conjecture, due to Zudilin [55], on certain rational
approximations of *\zeta*(4). The proofs are
based on a hypergeometric identity between a single sum and a multiple
sum due to Andrews [3].
We hope that it will be possible to apply our construction
to the more general linear forms constructed by Zudilin [56], with the
ultimate goal of strengthening his result that one of
the numbers *\zeta*(5),*\zeta*(7),*\zeta*(9),*\zeta*(11)
is irrational.

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