In 2000, Rivoal showed that infinitely many Riemann zeta values at odd integers were irrational, and even linearly independent over Q. To this end, he constructed explicit linear combinations of these numbers and applied Nesterenko's linear independence criterion (1985) to them.
In 2021, Fischler improved and generalized this result by constructing non-explicit linear combinations, with the help of a Thue-Siegel lemma. Nesterenko's criterion doesn't apply anymore, and we have to consider a new linear independence criterion based on Siegel's ideas.
We will present these two results and the major ideas involved in their proofs. Then, we will state a generalization of Fischler's theorem, about linear independence of values of Dirichlet L-functions. This result also holds outside of the disk of convergence of the L-functions, by working with their analytic continuation.
