Bibliographic record
Abstract
Let E be an elliptic curve defined over the rational numbers and r a fixed integer. Using a probabilistic model consistent with the Chebotarev density theorem for the division fields of E and the Sato–Tate distribution, Lang and Trotter conjectured an asymptotic formula for the number of primes up to x which have Frobenius trace equal to r, where r is a fixed integer. However, as shown in this note, this asymptotic estimate cannot hold for allr in the interval with a uniform bound for the error term, because an estimate of this kind would contradict the Chebotarev density theorem as well as the Sato–Tate conjecture. The purpose of this note is to refine the Lang–Trotter conjecture, by taking into account the “semicircular law,” to an asymptotic formula that conjecturally holds for arbitrary integers r in the interval , with a uniform error term. We demonstrate consistency of our refinement with the Chebotarev density theorem for a fixed division field, and with the Sato–Tate conjecture. We also present numerical evidence for the refined conjecture.
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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 | 0.007 | 0.049 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.006 |
| Scholarly communication | 0.003 | 0.011 |
| Open science | 0.003 | 0.003 |
| Research integrity | 0.002 | 0.004 |
| 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".