Shidlovsky's multiplicity estimate and Irrationality of zeta values

Abstract : 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.
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https://hal.archives-ouvertes.fr/hal-01374286
Contributor : Stéphane Fischler <>
Submitted on : Friday, September 30, 2016 - 11:07:12 AM
Last modification on : Thursday, January 11, 2018 - 6:12:19 AM
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  • HAL Id : hal-01374286, version 1
  • ARXIV : 1609.09770

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Stéphane Fischler. Shidlovsky's multiplicity estimate and Irrationality of zeta values. 2016. ⟨hal-01374286⟩

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