Bibliographic record
Abstract
The Hilbert-Pólya Conjecture supposes that there exists an operator in a Hilbert space whose eigenvalues are the zeroes of the Riemann Zeta function ζ(s). This conjecture, if true, would very likely expedite the proof of the Riemann Hypothesis, namely that the non-trivial zeroes of ζ(s) have real part 1/2. In this thesis we summarize work by Berry, Keating and others in constructing such an operator. Although the work so far has not yet yielded such an operator, some have been found that have properties very close to what is desired. We also summarize a (partially proven) conjecture by Montgomery that motivates the search for this operator. He conjectures that the pair correlation function for the spacing between the imaginary parts of the Riemann zeroes is the same as the correlation function for the spacing between eigenvalues of random Gaussian unitary matrices.
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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 | 0.008 | 0.040 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.002 | 0.003 |
| Science and technology studies | 0.004 | 0.013 |
| Scholarly communication | 0.005 | 0.015 |
| Open science | 0.002 | 0.005 |
| Research integrity | 0.003 | 0.006 |
| Insufficient payload (model declined to judge) | 0.011 | 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; 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".