COMPLEX MOMENTS AND THE DISTRIBUTION OF VALUES OF OVER FUNCTION FIELDS WITH APPLICATIONS TO CLASS NUMBERS
Bibliographic record
Abstract
In this paper we investigate the moments and the distribution of , where varies over quadratic characters associated to square-free polynomials of degree over , as . Our first result gives asymptotic formulas for the complex moments of in a large uniform range. Previously, only the first moment has been computed due to the work of Andrade and Jung. Using our asymptotic formulas together with the saddle-point method, we show that the distribution function of is very close to that of a corresponding probabilistic model. In particular, we uncover an interesting feature in the distribution of large (and small) values of , which is not present in the number field setting. We also obtain -results for the extreme values of , which we conjecture to be the best possible. Specializing and making use of one case of Artin's class number formula, we obtain similar results for the class number associated to . Similarly, specializing to we can appeal to the second case of Artin's class number formula and deduce analogous results for , where is the regulator of .
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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 | 0.003 | 0.022 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.003 | 0.002 |
| Science and technology studies | 0.001 | 0.004 |
| Scholarly communication | 0.002 | 0.005 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.004 | 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 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".