Oscillation results for the summatory functions of fake $\mu $’s
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Bibliographic record
Abstract
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.
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Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.003 | 0.004 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it