This material has been published in
Mem. Amer. Math. Soc. (to appear),
the only definitive repository of the content that has been
certified and accepted after peer review. Copyright and all rights therein
are retained by the American Mathematical Society.
This material may not be copied or reposted
without explicit permission.
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+2h),
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.
The following versions are available:
Back to Christian Krattenthaler's
home page.