Biases in the Shanks—Rényi Prime Number Race
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Bibliographic record
Abstract
Rubinstein and Sarnak investigated systemsof inequalities of the form π(x; q, a1) > … > π(x; q, ar), where p(x; q, b) denotes the number of primes up to x that are congruent to b mod q. They showed, under standard hypotheses on the zeros of Dirichlet L-functions mod q, that the set of positive real numbers x for which these inequalities hold has positive (logarithmic) density δq;al, … .ar > 0. They also discovered the surprising fact that a certain distribution associated with these densities is not symmetric under permutations of the residue classes aj in general, even if the aj are all squaresor all nonsquares mod q (a condition necessary to avoid obvious biases of the type first observed by Chebyshev). This asymmetry suggests, contrary to prior expectations, that the densities δq;al , …,ar themselves vary under permutations of the aj. Here we derive (under the hypotheses used by Rubinstein and Sarnak) a general formula for the densities δq;al , …,ar, and We use this formula to calculate many of these densities when q ≤ 12 and r ≤ 4. For the special moduli q = 8 and q = 12, and for {al, a2,a3} a permutation of the nonsquares {3, 5, 7} mod 8 and {5, 7, 11} mod 12, respectively, we rigorously bound the error in our calculations, thus verifying that these densities are indeed asymmetric under permutation of the aj. We also determine several situations in which the densities δq;al , …, ar remain unchanged under certain permutations of the aj, and some situations in which they are provably different.
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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 | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.016 | 0.003 |
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 it