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SHIDLOVSKY’S MULTIPLICITY ESTIMATE AND IRRATIONALITY OF ZETA VALUES

Part of: Zeta and $L$-functions: analytic theory Diophantine approximation, transcendental number theory Differential equations in the complex domain

Published online by Cambridge University Press:  18 June 2018

STÉPHANE FISCHLER
STÉPHANE FISCHLER*
Affiliation:
Laboratoire de Mathématiques d’Orsay, Univ. Paris-Sud, CNRS, Université Paris-Saclay, 91405 Orsay, France email stephane.fischler@math.u-psud.fr
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Abstract

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In this paper we follow the approach of Bertrand–Beukers (and of Bertrand’s later work), 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 (that is, 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 the 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.

Keywords

irrationalityzeta valuesPadé approximationShidlovsky’s lemma

MSC classification

Primary: 11J72: Irrationality; linear independence over a field
Secondary: 11M06: $zeta (s)$ and $L(s, chi)$ 34M03: Linear equations and systems
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 105 , Issue 2 , October 2018 , pp. 145 - 172
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

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