Subword Complexity and (non)-automaticity of certain completely multiplicative functions

Abstract

In this article, we prove that for a completely multiplicative function $f$ from $\mathbb{N}^*$ to a field $K$ such that the set $$\{p \;|\; f(p)\neq 1_K \;\mbox{and }p \mbox{ is prime}\}$$ is finite, the asymptotic subword complexity of $f$ is $\Theta(n^t)$, where $t$ is the number of primes $p$ that $f(p)\neq 0_K, 1_K$. This proves in particular that sequences like $((-1)^{v_2(n)+v_3(n)})_n$ are not $k$-automatic for $k\geq 2$.