  In this paper we follow the approach of Bertrand-Beukers (and of later work
of Bertrand), based on differential Galois theory, to prove a very general
version of Shidlovsky's lemma that applies to Pad{\'e} approximation problems
at several points, both at functional and numerical levels (i.e., before and
after evaluating at a specific point). This allows us to obtain a new proof of
the Ball-Rivoal theorem on irrationality of infinitely many values of Riemann
zeta function at odd integers, inspired by the proof of the Siegel-Shidlovsky
theorem on values of E-functions: Shidlovsky's lemma is used to replace
Nesterenko's linear independence criterion with Siegel's, so that no lower
bound is needed on the linear forms in zeta values. The same strategy provides
a new proof, and a refinement, of Nishimoto's theorem on values of L-functions
of Dirichlet characters.
