English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

In this paper we follow the approach of Bertrand-Beukers (and of later workof Bertrand), based on differential Galois theory, to prove a very generalversion of Shidlovsky's lemma that applies to Pad{\'e} approximation problemsat several points, both at functional and numerical levels (i.e., before andafter evaluating at a specific point). This allows us to obtain a new proof ofthe Ball-Rivoal theorem on irrationality of infinitely many values of Riemannzeta function at odd integers, inspired by the proof of the Siegel-Shidlovskytheorem on values of E-functions: Shidlovsky's lemma is used to replaceNesterenko's linear independence criterion with Siegel's, so that no lowerbound is needed on the linear forms in zeta values. The same strategy providesa new proof, and a refinement, of Nishimoto's theorem on values of L-functionsof Dirichlet characters.

Shidlovsky's multiplicity estimate and Irrationality of zeta values