Schanuel's conjecture and the exceptional set of $\gamma$-th arithmetic zeta functions

TL;DR

This paper investigates the arithmetic properties of numbers of the form n^γ, explores a related conjecture, and demonstrates its dependence on Schanuel's conjecture, contributing to the understanding of transcendental number theory.

Contribution

It introduces a new conjecture related to the arithmetic nature of n^γ and shows its implication from Schanuel's conjecture, linking these concepts in transcendence theory.

Findings

The conjecture is a consequence of Schanuel's conjecture.

Provides insights into the arithmetic nature of exponential numbers.

Establishes connections between conjectures in transcendence theory.

Abstract

In this work, we study the arithmetic nature of the numbers of the form $n^{\g}$, for $n \in \N$ and $\g\in \C$. We also consider a related conjecture and we show that it is a consequence of the unipresent Schanuel's conjecture.