Pré-Publication, Document De Travail Année : 2023

Rational approximations to values of $E$-functions

Résumé

We solve a long standing problem in the theory of Siegel's $E$-functions, initiated by Lang for Bessel's function $J_0$ 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 $J_0$ 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.

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Dates et versions

hal-04349259 , version 1 (17-12-2023)
hal-04349259 , version 2 (10-01-2024)
hal-04349259 , version 3 (08-07-2025)

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  • HAL Id : hal-04349259 , version 3

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Stéphane Fischler, Tanguy Rivoal. Rational approximations to values of $E$-functions. 2025. ⟨hal-04349259v3⟩
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