# On multiplicative automatic sequences

## TL;DR

This paper characterizes automatic multiplicative sequences, showing they are either Dirichlet characters or zero outside small primes, resolving a question in the field.

## Contribution

It proves a classification theorem for automatic multiplicative sequences, answering an open question and providing a clear dichotomy.

## Key findings

- Automatic multiplicative sequences are either Dirichlet characters or zero outside small primes.
- The result confirms a conjecture posed by Bell, Bruin, and Coons.
- Similar results were independently obtained by Klurman and Kurlberg.

## Abstract

We show that any automatic multiplicative sequence either coincides with a Dirichlet character or is identically zero when restricted to integers not divisible by small primes. This answers a question of Bell, Bruin and Coons. A similar result was obtained independently by Klurman and Kurlberg.