Oscillation results for the summatory functions of fake mu's

TL;DR

This paper extends oscillation results for the summatory functions of fake 's, a class of multiplicative functions, at various scales, generalizing previous findings and connecting to powerfree and powerful numbers.

Contribution

It establishes new oscillation results for all nontrivial fake 's at scales determined by a critical index, generalizing prior work and unifying results for related number-theoretic functions.

Findings

Oscillation results at scales x^{1/2} for all nontrivial fake 's.

Recovery of oscillation results for powerfree and powerful numbers.

Generalization of techniques to a broader class of multiplicative functions.

Abstract

Mossinghoff, Trudgian, and the first author~\cite{MMT23} 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 paper, 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~\cite{MMT23}. 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.