Bibliographic record
Abstract
For various arithmetic functions f : N ? R, the behavior of f(n!) and that of ? n?N f(n!) can be intriguing. For instance, for some functions f, we have f(n!) = ? k?n f(k), for others, we have f(n!) = ? p?n f(p) (where the sum runs over all the primes p ? n). Also, for some f, their minimum order coincides with lim n? ?f(n!), for others, it is their maximum order that does so. Here, we elucidate such phenomena and more generally, we embark on a study of f(n!) and of ? n?N f(n!) for a wide variety of arithmetical functions f. In particular, letting d(n) and ?(n) stand respectively for the number of positive divisors of n and the sum of the positive divisors of n, we obtain new accurate asymptotic expansions for d(n!) and ? (n!). Furthermore, setting ? 1(n) := max{d | n : d ? ?n} and observing that no one has yet obtained an asymptotic value for ? n?N ? 1(n) as N ? ?, we show how one can obtain the asymptotic value of ? n?N ?1(n!).
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.002 | 0.002 |
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; both teacher heads agree on what is shown here.
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".