Moments of the derivative of the characteristic polynomial of unitary matrices
Bibliographic record
Abstract
Let [Formula: see text] be the characteristic polynomial of a Haar distributed unitary matrix [Formula: see text]. It is believed that the distribution of values of [Formula: see text] model the distribution of values of the Riemann zeta-function [Formula: see text]. This principle motivates many avenues of study. Of particular interest is the behavior of [Formula: see text] and the distribution of its zeros (all of which lie inside or on the unit circle). In this paper, we present several identities for the moments of [Formula: see text] averaged over [Formula: see text], for [Formula: see text] as well as specialized to [Formula: see text]. Additionally, we prove, for positive integer [Formula: see text], that the polynomial [Formula: see text] of degree [Formula: see text] in [Formula: see text] divides the polynomial [Formula: see text] which is of degree [Formula: see text] in [Formula: see text] and that the ratio, [Formula: see text], of these moments factors into linear factors modulo [Formula: see text] if [Formula: see text] is prime. We also discuss the relationship of these moments to a solution of a second-order nonlinear Painléve differential equation. Finally we give some formulas in terms of the [Formula: see text] hypergeometric series for the moments in the simplest case when [Formula: see text], and also study the radial distribution of the zeros of [Formula: see text] in that case.
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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 | 0.001 | 0.008 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.006 | 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 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".