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

TL;DR

This paper proves that certain multiplicative functions related to the Möbius function over squarefree and cubefree integers have unbounded partial sums, extending the understanding of the Erdős discrepancy problem in these contexts.

Contribution

It establishes unboundedness of partial sums for multiplicative functions involving Möbius and indicator functions over squarefree and cubefree integers, using advanced analytical methods.

Findings

Unbounded partial sums for =^2g and =_2^2g.

Extension of Erd
53s discrepancy problem to specific multiplicative functions.

Application of Klurman, Mangerel, and Tao's methods to new classes of functions.

Abstract

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 $f=\mu^2g$ and $f=\mu_2^2g$ 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 Erd\H{o}s discrepancy problem.