There are infinitely many irrational values of the Riemann zeta function at odd integers

We provide a lower bound for the dimension of the vector space spanned over the rationals by 1 and by the values of the Riemann zeta function at the f irst n odd integers. We prove that this dimension increases at least like a constant times log(n). As a consequence, the zeta function takes infinitel y many irrational values at odd integers. (C) 2000 Academie des sciences/Ed itions scientifiques et medicales Elsevier SAS.