Shidlovsky's multiplicity estimate and Irrationality of zeta values

TL;DR

This paper extends Shidlovsky's lemma using differential Galois theory to prove the irrationality of infinitely many odd zeta values, offering new proofs and refinements of key theorems in number theory.

Contribution

It generalizes Shidlovsky's lemma to multiple points and applies it to provide novel proofs of the irrationality of zeta and L-function values, removing previous limitations.

Findings

New proof of the Ball-Rivoal theorem on zeta values

Refinement of Nishimoto's theorem on L-functions

Elimination of lower bounds in linear independence proofs

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 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.