On Correlations of Liouville-like Functions

Let $\mathcal{A}$ be a set of mutually coprime positive integers, satisfying \begin{align*} \sum\limits_{a\in$\mathcal{A}$}\frac{1}{a} = \infty. \end{align*} Define the (possibly non-multiplicative) “Liouville-like” functions \begin{align*} λ_{$\mathcal{A}$}(n) = (-1)^{#{a:a|n, a \in $\mathcal{A}$}} $\text{ or }$ (-1)^{#{a:a^ν\parallel n, a \in $\mathcal{A}$, ν\in $\mathbb{N}$}}. \end{align*} We show that \begin{align*} \lim\limits_{x\to\infty}\frac{1}{x}\sum\limits_{n \leq x} λ_$\mathcal{A}$(n) = 0 \end{align*} holds, answering a question of de la Rue.