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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2506.15119 |
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Table of Contents:
- We prove that some holomorphic continuations of functions in the classes $\mathbf{an}^*$ and $\mathcal{G}$ are definable in the o-minimal structures $\mathbb{R}_{\mathrm{an}^*}$ and $\mathbb{R}_{\mathcal{G}}$ respectively. More specifically, we give complex domains on which the holomorphic continuations are definable, and show they are optimal. As an application, we describe optimal domains on which the Riemann $ζ$ function is definable in o-minimal expansions of $\mathbb{R}_{\mathrm{an}^*,\exp}$ and on which the $Γ$ function is definable in o-minimal expansions of $\mathbb{R}_{\mathcal{G},\exp}$.