Definability of complex functions in o-minimal structures

TL;DR

This paper demonstrates that certain complex functions, including the Riemann zeta and gamma functions, are definable within specific o-minimal structures on optimal complex domains, advancing the understanding of their logical complexity.

Contribution

It establishes the definability of holomorphic continuations of functions in classes n* and alg in corresponding o-minimal structures and identifies optimal domains for these definability results.

Findings

Holomorphic continuations are definable on optimal complex domains.

Riemann ta function is definable in n*-expansion.

Gamma function is definable in alg-expansion.

Abstract

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 $\zeta$ function is definable in o-minimal expansions of $\mathbb{R}_{\mathrm{an}^*,\exp}$ and on which the $\Gamma$ function is definable in o-minimal expansions of $\mathbb{R}_{\mathcal{G},\exp}$.