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arXiv · 2312.12043

Rational approximations to values of $E$-functions

Stéphane Fischler·Tanguy Rivoal
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Abstract

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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Stéphane Fischler, Tanguy Rivoal. 2023-12-19. Rational approximations to values of $E$-functions. https://arxiv.org/abs/2312.12043

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