On the proximity of multiplicative functions to the number of distinct prime factors function
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Bibliographic record
Abstract
Abstract Given an additive function f and a multiplicative function g , let E ( f , g ; x ) = #{ n ≤ x : f ( n ) = g ( n )}. We study the size of E ( ω , g ; x ) and E (Ω, g ; x ), where ω ( n ) stands for the number of distinct prime factors of n and Ω( n ) stands for the number of prime factors of n counting multiplicity. In particular, we show that E ( ω , g ; x ) and E (Ω, g ; x ) are <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mtable> <m:mtr> <m:mtd> <m:mstyle> <m:mi>O</m:mi> <m:mfenced> <m:mfrac> <m:mi>x</m:mi> <m:msqrt> <m:mi>log</m:mi> <m:mi>log</m:mi> <m:mo></m:mo> <m:mi>x</m:mi> </m:msqrt> </m:mfrac> </m:mfenced> </m:mstyle> </m:mtd> </m:mtr> </m:mtable> </m:math> $\begin{array}{} \displaystyle O\left(\frac{x}{\sqrt{\log\log x}}\right) \end{array}$ for any integer valued multiplicative function g . This improves an earlier result of De Koninck, Doyon and Letendre.
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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 | 0.004 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.002 | 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 it