On Arratia’s coupling and the Dirichlet law for the factors of a random integer
Bibliographic record
Abstract
Let x ≥ 2 , let N x be an integer chosen uniformly at random from the set ℤ ∩ [ 1 , x ] , and let ( V 1 , V 2 , ... ) be a Poisson–Dirichlet process of parameter 1 . We prove that there exists a coupling of these two random objects such that 𝔼 ∑ i ≥ 1 | log P i - V i log x | ≍ 1 , where the implied constants are absolute and N x = P 1 P 2 ⋯ is the unique factorization of N x into primes or ones with the P i ’s being non-increasing. This establishes a 2002 conjecture of Arratia, who constructed a coupling for which the left-hand side in the above estimate is ≪ log log x , and who also proved that the left-hand side is ≥ 1 - o ( 1 ) for all couplings. In addition, we use our refined coupling to give a probabilistic proof of the Dirichlet law for the average distribution of the integer factorization into k parts proved in 2023 by Leung and we improve on its error term.
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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 | 0.007 | 0.042 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.003 |
| Bibliometrics | 0.003 | 0.002 |
| Science and technology studies | 0.003 | 0.008 |
| Scholarly communication | 0.005 | 0.009 |
| Open science | 0.003 | 0.004 |
| Research integrity | 0.003 | 0.006 |
| Insufficient payload (model declined to judge) | 0.016 | 0.004 |
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".