Abstract:
It is shown that values of $G$-functions satisfying a system of linear differential equations are irrational at rational points $a/b$ with $a\in\mathbb Z$ and $b\in\mathbb N$ such that
$b>C(\varepsilon)|a|^{2+\varepsilon}$ for an arbitrary positive $\varepsilon$. In the case of a generalized polylogarithmic function
$$
f(z)=\sum_{\nu=1}^\infty\frac{z^\nu}{(\nu+\lambda)^m}, \quad m\geqslant 2, \enskip \lambda\in\mathbb Q\setminus\{-1,-2,\dots\},
$$
an explicit form of $C(\varepsilon)$ is found.
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