Irrationality of special values of formal Laurent series represented by the formal Mellin transform of $G$-functions

Abstract

Let $p$ be a prime number and ${\mathbb{C}}_p$ the completion of algebraic closure of ${\mathbb{Q}}_p$. Let $K$ be an algebraic number field. We fix an embedding ${\iota }_p:\overline{\mathbb{Q}}\hookrightarrow {\mathbb{C}}_p$ and denote $K_p$ the completion of $K$ with respect to the embedding ${\iota }_p$. Let $g(z)\in K[[z]]$ and denote by $\mathcal{M}(g)(z)\in \frac{1}{z}K\left[\left[\frac{1}{z}\right]\right]$ the formal Mellin transform of $g(z)$. In this article, we prove that if $\mathcal{M}(g)(z)$ has a good Padé approximation, the special values $\mathcal{M}(g)(\alpha )$ are convergent in $K_p$ and irrational for infinitely many $\alpha \in \mathbb{Q}\cap \left({\mathbb{Q}}_p\setminus {\mathbb{Z}}_p\right)$ satisfying certain conditions. This result can be regarded as a partial generalization of the method of Beukers in his proof the irrationality of special values of $p$-adic Hurwitz zeta functions.

arXiv commentAlthough we state that Corollary4.4(see page 12) does hold, we do not know whether the claim holds in general. So the proof of Theorem1.2 does not work