On the values of $G$-functions

Abstract

In this paper we study the set $\mathbf{G}$ of values at algebraic points of analytic continuations of $G$-functions (in the sense of Siegel). This subring of $\mathbb{C}$ contains values of elliptic integrals, multiple zeta values, and values at algebraic points of generalized hypergeometric functions $_{p+1}F_p$ with rational coefficients. Its group of units contains non-zero algebraic numbers, $\pi$, $\Gamma(a/b)^b$ and $B(x,y)$ (with $a,b\in \mathbb{Z}$ such that $a/b\not\in \mathbb Z$, and $x,y\in\mathbb{Q}$ such that $B(x,y)$ exists and is non-zero). We prove that for any $\xi \in \mathbf{G}$, both $\operatorname{Re} \xi$ and $\mathrm{Im} \, \xi$ can be written as $f(1)$, where $f$ is a $G$-function with rational coefficients of which the radius of convergence can be made arbitrarily large. As an application, we prove that quotients of elements of $\mathbf{G} \cap \mathbb{R}$ are exactly the numbers which can be written as limits of sequences $a_n/b_n$, where $\sum_{n=0}^{\infty} a_n z^n$ and $\sum_{n=0}^{\infty} b_n z^n$ are $G$-functions with rational coefficients. This result provides a general setting for irrationality proofs in the style of Apéry for $\zeta(3)$, and gives answers to questions asked by T. Rivoal in “Approximations rationnelles des valeurs de la fonction Gamma aux rationnels: le cas des puissances”, Acta Arith. 142 (2010), no. 4, 347–365.