Oscillation results for the summatory functions of fake mu's

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 $({\epsilon }_j{)}_{j=1}^{\infty }$ such that $f(p^j)={\epsilon }_j$ 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 ${\epsilon }_1=-1$ and ${\epsilon }_2=1$. 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.

39 pages, revised based on referee comments

References in corpus (1)

• An annotated bibliography for comparative prime number theory