2023/12/19 by Stéphane Fischler, Fischler, Stéphane, Tanguy Rivoal +1
Mathematics · #11J82 (Primary) #11J91 (Secondary) #Advanced Algebra and Geometry #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2312.12043
openalex publication_date 2023/12/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We solve a long standing problem in the theory of Siegel's E-functions, initiated by Lang for Bessel's function J0 in the 60's and considered in full generality by G. Chudnovsky in the 80's: we prove that irrational values taken at rational points by E-functions with rational Taylor coefficients have irrationality exponent equal to 2. This result had been obtained before by Zudilin under strong assumptions on algebraic independence of E-functions, satisfied by J0 but not by all hypergeometric E-functions for instance. We remove them using a new generalization of Shidlovskii's lemma, analogous to zero estimates on commutative algebraic groups in which obstructions come from algebraic subgroups.