# A note on multiplicative automatic sequences, II

## TL;DR

This paper proves that multiplicative automatic functions are either closely related to Dirichlet characters or become zero on large primes, confirming a conjecture in the field.

## Contribution

It establishes a definitive characterization of q-automatic multiplicative functions, resolving a conjecture about their structure.

## Key findings

- Such functions either match a Dirichlet character or vanish on large primes.
- The result confirms a strong form of a conjecture by Bell, Bruin, and Coons.
- Provides a complete classification of multiplicative automatic functions.

## Abstract

We prove that any $q$-automatic multiplicative function $f:\mathbb{N}\to\mathbb{C}$ either essentially coincides with a Dirichlet character, or vanishes on all sufficiently large primes. This confirms a strong form of a conjecture of J. Bell, N. Bruin, and M. Coons.