# A note on multiplicative automatic sequences

## TL;DR

This paper proves that all completely multiplicative functions that are automatic are essentially Dirichlet characters, confirming a conjecture and answering a question in the intersection of automata theory and number theory.

## Contribution

It establishes that q-automatic completely multiplicative functions must coincide with Dirichlet characters, providing a significant link between automata and multiplicative number theory.

## Key findings

- Any q-automatic completely multiplicative function is a Dirichlet character.
- Confirms a conjecture by Bell, Bruin, and Coons.
- Answers a question posed by Allouche and Goldmakher.

## Abstract

We prove that any $q$-automatic completely multiplicative function $f:\mathbb{N}\to\mathbb{C}$ essentially coincides with a Dirichlet character. This answers a question of J. P. Allouche and L. Goldmakher and confirms a conjecture of J. Bell, N. Bruin and M. Coons for completely multiplicative functions. Further, assuming two standard conjectures in number theory, the methods allows for removing the assumption of completeness.