Oscillation results for the summatory functions of fake mu's
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915458420047872 |
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| author | Martin, Greg Yip, Chi Hoi |
| author_facet | Martin, Greg Yip, Chi Hoi |
| contents | Mossinghoff, Trudgian, and the first author~\cite{MMT23} 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 $(\varepsilon_j)_{j=1}^{\infty}$ such that $f(p^j) = \varepsilon_j$ for all primes $p$. They investigated comparative number-theoretic results for fake $μ$'s and in particular proved oscillation results at scale $\sqrt{x}$ for the summatory functions of fake $μ$'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 $μ$'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 $μ$'s. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_06610 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Oscillation results for the summatory functions of fake mu's Martin, Greg Yip, Chi Hoi Number Theory Primary 11N37, Secondary 11A25, 11M06, 11M26 Mossinghoff, Trudgian, and the first author~\cite{MMT23} 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 $(\varepsilon_j)_{j=1}^{\infty}$ such that $f(p^j) = \varepsilon_j$ for all primes $p$. They investigated comparative number-theoretic results for fake $μ$'s and in particular proved oscillation results at scale $\sqrt{x}$ for the summatory functions of fake $μ$'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 $μ$'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 $μ$'s. |
| title | Oscillation results for the summatory functions of fake mu's |
| topic | Number Theory Primary 11N37, Secondary 11A25, 11M06, 11M26 |
| url | https://arxiv.org/abs/2411.06610 |