Bibliographic record
Abstract
The irregular distribution of prime numbers amongst the integers has found multiple uses, from engineering applications of cryptography to quantum theory. The degree to which this distribution can be predicted thus has become a subject of current interest. Here, we present a computational analysis of the deviations between the actual positions of the prime numbers and their predicted positions from Riemann’s counting formula, focused on the variance function of these deviations from sequential enumerative bins. We show empirically that these deviations can be described by a class of probabilistic models known as the Tweedie exponential dispersion models that are characterized by a power law relationship between the variance and the mean, known by biologists as Taylor’s power law and by engineers as fluctuation scaling. This power law behavior of the prime number deviations is remarkable in that the same behavior has been found within the distribution of genes and single nucleotide polymorphisms (SNPs) within the human genome, the distribution of animals and plants within their habitats, as well as within many other biological and physical processes. We explain the common features of this behavior through a statistical convergence effect related to the central limit theorem that also generates 1/f noise.
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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 | 0.013 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.003 | 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".