Bibliographic record
Abstract
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.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.003 | 0.021 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.003 | 0.001 |
| Science and technology studies | 0.002 | 0.005 |
| Scholarly communication | 0.002 | 0.005 |
| Open science | 0.002 | 0.003 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.006 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".