Bibliographic record
Abstract
Negative correlations in the distribution of prime numbers are found to display a scale invariance. This occurs in conjunction with a nonstationary behavior. We compare the prime number series to a type of fractional Brownian motion which incorporates both the scale invariance and the nonstationary behavior. Interesting discrepancies remain. The scale invariance also appears to imply the Riemann hypothesis and we study the use of the former as a test of the latter. 1 Scale Invariance The distribution of prime numbers has been a source of fascination for mathematicians for centuries. Of interest to physicists is the fact that the distribution of prime numbers exhibits certain characteristics that are usually associated with chaotic and complex systems. The data on the prime number series is plentiful and a surprisingly rich structure is uncovered when this data is probed in various ways [1]. The closely associated zeros of the Riemann zeta function have been studied even more intensely from a physics viewpoint [2]. We shall show that prime number differences display a scale invariance in their correlations. The differences also display a nonstationary behavior that coexists with the scale invariance. We shall identify a random process, fractional Brownian motion of a lesser known type, that displays these and other properties of the prime number series. But there are still quantitative differences which suggest that a further generalization of fractional Brownian motion remains to be developed. The resolution of this puzzle could have practical value for descriptions of complex systems in physics and economics. In addition we shall describe how a breakdown of the Riemann hypothesis would not be compatible with the preservation of scale invariance.
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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) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 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 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".