Shidlovsky's multiplicity estimate and Irrationality of zeta values

Stéphane Fischler (1)

1 LM-Orsay - Laboratoire de Mathématiques d'Orsay

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 Padé 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.

Domaines: Théorie des nombres [math.NT]

Stéphane Fischler. Shidlovsky's multiplicity estimate and Irrationality of zeta values. Journal of the Australian Mathematical Society, 2018, 105 (2), pp.145-172. ⟨hal-01374286⟩