A note on multiplicative automatic sequences

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

A note on multiplicative automatic sequences, II

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