The Erdős discrepancy problem over the squarefree and cubefree integers
Let g:N→{−1,1}$g:\mathbb {N}\rightarrow \lbrace -1,1\rbrace$ be a completely multiplicative function, μ$\mu$ be the Möbius function and μ22(n)$\mu _2^2(n)$ be the indicator that n$n$ is cubefree. We prove that f=μ2g$f=\mu ^2g$ and f=μ22g$f=\mu _2^2g$ 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 Erdős discrepancy problem.