The Erdős discrepancy problem over the squarefree and cubefree integers

Abstract

Let $g:\mathbb{N}\to \{-1,1\}$ be a completely multiplicative function, $\mu$ be the Möbius 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 Erdős discrepancy problem.

24 pages. The conjecture from version 1 is now a Theorem. To appear in Mathematika

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• Multiplicative functions that are close to their mean