The Erd\H{o}s discrepancy problem over the squarefree and cubefree integers

Let $g:\mathbb{N}\to \{-1,1\}$ be a completely multiplicative function, $\mu$ be the M"obius function and ${\mu }_2^2(n)$ be the indicator that $n$ is cubefree. We prove that $f={\mu }^{2}g$ and $f={\mu }_2^{2}g$ have unbounded partial sums. Our proofs are built upon Klurman and Mangerel’s proof of Chudakov’s conjecture, Klurman’s work on correlations of multiplicative functions and Tao’s resolution of the Erd\H{o}s discrepancy problem.