Inequities in the Shanks-Renyi Prime Number Race: An asymptotic formula\n for the densities
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Bibliographic record
Abstract
Chebyshev was the first to observe a bias in the distribution of primes in\nresidue classes. The general phenomenon is that if $a$ is a nonsquare\\mod q and\n$b$ is a square\\mod q, then there tend to be more primes congruent to $a\\mod q$\nthan $b\\mod q$ in initial intervals of the positive integers; more succinctly,\nthere is a tendency for $\\pi(x;q,a)$ to exceed $\\pi(x;q,b)$. Rubinstein and\nSarnak defined $\\delta(q;a,b)$ to be the logarithmic density of the set of\npositive real numbers $x$ for which this inequality holds; intuitively,\n$\\delta(q;a,b)$ is the "probability" that $\\pi(x;q,a) > \\pi(x;q,b)$ when $x$ is\n"chosen randomly". In this paper, we establish an asymptotic series for\n$\\delta(q;a,b)$ that can be instantiated with an error term smaller than any\nnegative power of $q$. This asymptotic formula is written in terms of a\nvariance $V(q;a,b)$ that is originally defined as an infinite sum over all\nnontrivial zeros of Dirichlet $L$-functions corresponding to characters\\mod q;\nwe show how $V(q;a,b)$ can be evaluated exactly as a finite expression. In\naddition to providing the exact rate at which $\\delta(q;a,b)$ converges to\n$\\frac12$ as $q$ grows, these evaluations allow us to compare the various\ndensity values $\\delta(q;a,b)$ as $a$ and $b$ vary modulo $q$; by analyzing the\nresulting formulas, we can explain and predict which of these densities will be\nlarger or smaller, based on arithmetic properties of the residue classes $a$\nand $b\\mod q$. For example, we show that if $a$ is a prime power and $a'$ is\nnot, then $\\delta(q;a,1) < \\delta(q;a',1)$ for all but finitely many moduli $q$\nfor which both $a$ and $a'$ are nonsquares. Finally, we establish rigorous\nnumerical bounds for these densities $\\delta(q;a,b)$ and report on extensive\ncalculations of them.\n
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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 | 0.003 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.002 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.001 | 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 it