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In this paper we generalize Nesterenko's criterion to the case where thesmall linear forms have an oscillating behaviour (for instance given by thesaddle point method). This criterion provides both a lower bound for thedimension of the vector space spanned over the rationals by a family of realnumbers, and a measure of simultaneous approximation to these numbers (namely,an upper bound for the irrationality exponent if 1 and only one other numberare involved). As an application, we prove an explicit measure of simultaneousapproximation to $\zeta(5)$, $\zeta(7)$, $\zeta(9)$, and $\zeta(11)$, usingZudilin's proof that at least one of these numbers is irrational.

Nesterenko's criterion when the small linear forms oscillate