On Arratia’s coupling and the Dirichlet law for the factors of a random integer
Bibliographic record
Abstract
Let <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> , let <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>N</mml:mi> <mml:mi>x</mml:mi> </mml:msub> </mml:math> be an integer chosen uniformly at random from the set <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ℤ</mml:mi> <mml:mo>∩</mml:mo> <mml:mo>[</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mi>x</mml:mi> <mml:mo>]</mml:mo> </mml:mrow> </mml:math> , and let <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>V</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>V</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>...</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> be a Poisson–Dirichlet process of parameter <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mn>1</mml:mn> </mml:math> . We prove that there exists a coupling of these two random objects such that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"> <mml:mrow> <mml:mi>𝔼</mml:mi> <mml:mspace width="0.166667em"/> <mml:munder> <mml:mo>∑</mml:mo> <mml:mrow> <mml:mi>i</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:munder> <mml:mo>|</mml:mo> <mml:mrow> <mml:mo form="prefix">log</mml:mo> <mml:msub> <mml:mi>P</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo>-</mml:mo> <mml:msub> <mml:mi>V</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo form="prefix">log</mml:mo> <mml:mi>x</mml:mi> </mml:mrow> <mml:mo>|</mml:mo> <mml:mo>≍</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> where the implied constants are absolute and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>N</mml:mi> <mml:mi>x</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>P</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:msub> <mml:mi>P</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>⋯</mml:mo> </mml:mrow> </mml:math> is the unique factorization of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>N</mml:mi> <mml:mi>x</mml:mi> </mml:msub> </mml:math> into primes or ones with the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>P</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:math> ’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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>≪</mml:mo> <mml:mo form="prefix">log</mml:mo> <mml:mspace width="-0.166667em"/> <mml:mo form="prefix">log</mml:mo> <mml:mi>x</mml:mi> </mml:mrow> </mml:math> , and who also proved that the left-hand side is <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>≥</mml:mo> <mml:mn>1</mml:mn> <mml:mo>-</mml:mo> <mml:mi>o</mml:mi> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>k</mml:mi> </mml:math> parts proved in 2023 by Leung and we improve on its error term.
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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 | 0.003 | 0.006 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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 teacher head, 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".