Fonction Z\^eta de Hurwitz p-adique et irrationalit\'e

TL;DR

This paper advances the understanding of p-adic zeta values' irrationality by leveraging Pade approximants of Lerch functions, leading to new results on their irrationality and linear independence.

Contribution

It introduces a novel approach using Pade approximants of Lerch functions in the p-adic setting to establish irrationality and linear independence of p-adic zeta values.

Findings

Proved irrationality of certain p-adic zeta values.

Established linear independence results for p-adic zeta values.

Extended methods from complex analysis to p-adic analysis.

Abstract

The knowledge on irrationality of p-adic zeta values has recently progressed. The irrationality of zeta_2(2), \zeta_2(3) and of a few other p-adic series of Dirichlet was obtained by F. Calegari. F. Beukers gave a more elementary proof of this result. In parallel, T. Rivoal has just obtained, in the complex case, some Pade approximants of Lerch functions. It is this work which, transposed to C_p, enables us to obtain results of irrationality and linear independence.