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

TL;DR

This paper investigates the automaticity of completely multiplicative sequences with finitely many non-trivial prime factors, proving constraints on their structure and the distribution of primes where the sequence differs from 1.

Contribution

It establishes that such sequences cannot be automatic if they have infinitely many primes with non-trivial values, and characterizes their form when decomposed with Dirichlet characters.

Findings

Automatic sequences with negligible non-trivial prime factors have finitely many such primes.

Sequences decomposed into a non-periodic automatic part and a Dirichlet character have at most one prime with non-trivial value.

The paper links automaticity to the distribution of prime factors in multiplicative sequences.

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 $0$.