(non)-automaticity of completely multiplicative sequences having negligible many non-trivial prime factors

Abstract

In this article we consider the completely multiplicative sequences $(a_n{)}_{n\in \mathbf{N}}$ defined on a field $\mathbf{K}$ and satisfying $$\sum _{p|p\le n,a_p\ne 1,p\in \mathbf{P}}\frac{1}{p}<\infty ,$$ where $\mathbf{P}$ is the set of prime numbers. We prove that if such sequences are automatic then they cannot have infinitely many prime numbers $p$ such that $a_p\ne 1$. Using this fact, we prove that if a completely multiplicative sequence $(a_n{)}_{n\in \mathbf{N}}$, vanishing or not, can be written in the form $a_n=b_n{\chi }_n$ such that $(b_n{)}_{n\in \mathbf{N}}$ is a non ultimately periodic, completely multiplicative automatic sequence satisfying the above condition, and $({\chi }_n{)}_{n\in \mathbf{N}}$ is a Dirichlet character or a constant sequence, then there exists only one prime number $p$ such that $b_p\ne 1$ or $0$.