Oscillation results for the summatory functions of fake $\mu $’s

Abstract Martin, Mossinghoff, and Trudgian [19] recently introduced a family of arithmetic functions called “fake $\mu $ ’s,” which are multiplicative functions for which there is a $\{-1,0,1\}$ -valued sequence $(\varepsilon _j)_{j=1}^{\infty }$ such that $f(p^j) = \varepsilon _j$ for all primes p . They investigated comparative number-theoretic results for fake $\mu $ ’s and, in particular, proved oscillation results at scale $\sqrt {x}$ for the summatory functions of fake $\mu $ ’s with $\varepsilon _1=-1$ and $\varepsilon _2=1$ . In this article, we establish new oscillation results for the summatory functions of all nontrivial fake $\mu $ ’s at scales $x^{1/2\ell }$ where $\ell $ is a positive integer (the “critical index”) depending on f ; for $\ell =1$ this recovers the oscillation results in [19]. Our work also recovers results on the indicator functions of powerfree and powerfull numbers; we generalize techniques applied to each of these examples to extend to all fake $\mu $ ’s.