(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 $$\sum_{p| p \leq n, a_p \neq 1, p \in \mathbf{P}}\frac{1}{p}<\infty,$$ 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 $a_{p}\neq 1$. 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 $b_p \neq 1$ or $0$.