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, C(m), £(m + 2),..., £(ra-f-2ft), 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 C(5), C(?) C(19) is irratio-
nal, and there exists an odd integer j between 5 and 165 such that 1, £(3) and ((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 C(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 £(5), C(7),C(9),£(11) *s
irrational.
Resume
Nous demontrons la « conjecture des denominateurs » du deuxieme auteur [40] sur
le denominateur commun des coefficients des combinaisons lineaires en les valeurs de
la fonction zeta de Riemann, recemment construites pour minorer la dimension de
l'espace vectoriel engendre sur Q par 1, ((m), £(m+2),..., £(ra-h2/i), ou m et h sont
des nombres entiers, m 2 et h 0. En particulier, comme corollaires immediats,
on obtient l'irrationalite d'au moins un des huit nombres £(5),£(7),... ,£(19) et
P existence d'un entier impair j entre 5 et 165 tel que 1, £(3) et £(j) sont lineairement
independants sur Q, ce qui ameliore des resultats de [41] et [8], respectivement.
Nous prouvons egalement une conjecture connexe, due a Vasilyev [49], ainsi qu'une
conjecture de Zudilin [55] portant sur certaines approximations rationnelles de C(4).
Les demonstrations sont basees sur une identite entre une somme simple et une
somme multiple, de nature hypergeometrique, due a Andrews [3]. Nous esperons
que notre construction pourra aussi etre appliquee aux combinaisons lineaires plus
Received by the editor March 3, 2005.
2000 Mathematics Subject Classification. Primary 11J72; Secondary 11J82, 33C20.
Key words and phrases. Irrationalite des valeurs de la fonction zeta de Riemann, series hyper-
geometriques.
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