2024/11/10 by Chi Hoi Yip, Martin, Greg, Yip, Chi Hoi
Mathematics · #11M06 #11M26 #FOS: Mathematics #Mathematical Analysis and Transform Methods #Number Theory (math.NT) #Primary 11N37 #Secondary 11A25 #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2411.06610
openalex publication_date 2024/11/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Mossinghoff, Trudgian, and the first author~\citeMMT23 recently introduced a family of arithmetic functions called ``fake μ's'', which are multiplicative functions for which there is a \-1,0,1\-valued sequence (εj)j=1∞ such that f(pj) = εj for all primes p. They investigated comparative number-theoretic results for fake μ's and in particular proved oscillation results at scale √(x) for the summatory functions of fake μ's with ε1=-1 and ε2=1. In this paper, we establish new oscillation results for the summatory functions of all nontrivial fake μ's at scales x1/2ℓ where ℓ is a positive integer (the ``critical index'') depending on f; for ℓ=1 this recovers the oscillation results in~\citeMMT23. 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 μ's.