Bibliographic record
Abstract
A Hadamard factorisation of the Riemann $ξ$-function is used to characterise the zeros of the zeta function through the theory of generalized gamma convolutions (GGC). Riemann's reciprocal $ξ$-function is expressed, for $α>1$, as the Laplace transform of a GGC, with an explicit Levy form and a reciprocal Thorin measure given by a sine-squared transform. The construction is then carried to the centre $α=\half$, where, through Riemann's original incomplete-gamma continuation of $ξ$, the primal law $ξ(\half+s)/ξ(\half)$ is shown to be a scale mixture of $Γ(2)$ densities. Two facts single out this representation: by the Steutel--Kristiansen theorem a positive mixture of $Γ(α)$ laws is infinitely divisible exactly when $α\in(0,2]$, so $Γ(2)$ is the threshold shape at which positivity of the mixing measure alone secures infinite divisibility; and that mixing measure $M$ is positive, which we prove cell by cell through an average-of-powers identity that turns an apparently signed alternating series into a monotone Leibniz series. The positive $M$ is the input the Thorin/GGC condition requires.
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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 | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.001 | 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 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".