On Correlations of Liouville-like Functions

Abstract

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*} \lambda_{\mathcal{A}}(n) = (-1)^{\#\{a:a|n, a \in \mathcal{A}\}} \text{ or } (-1)^{\#\{a:a^\nu\parallel n, a \in \mathcal{A}, \nu \in \mathbb{N}\}}. \end{align*} We show that \begin{align*} \lim\limits_{x\to\infty}\frac{1}{x}\sum\limits_{n \leq x} \lambda_\mathcal{A}(n) = 0 \end{align*} holds, answering a question of de la Rue.