Inequities in the Shanks-Renyi Prime Number Race: An asymptotic formula\n for the densities
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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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 | 0.008 | 0.052 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.003 | 0.001 |
| Science and technology studies | 0.002 | 0.008 |
| Scholarly communication | 0.003 | 0.009 |
| Open science | 0.003 | 0.005 |
| Research integrity | 0.001 | 0.006 |
| Insufficient payload (model declined to judge) | 0.008 | 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 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".