BOMBIERI–VINOGRADOV THEOREMS FOR MODULAR FORMS AND APPLICATIONS
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Bibliographic record
Abstract
In this article, we consider a prime number theorem for arithmetic progressions “weighted” by Fourier coefficients of modular forms, and we develop Siegel-Walfisz type and Bombieri–Vinogradov type estimates for such a modular analogue. As an application, we have a Turán type estimate for modular forms asserting that for any δ > 0 and non-CM normalised Hecke eigenform f, P f ( a , q ) ≤ q 2 + δ , with a possible exceptional set of q of density 0 (depending at most on f and δ), where ( a , q ) = 1 , P f ( a , q ) denotes the least prime p, with λ f ( p ) ≠ 0 , congruent to a ( mod q ) , and λ f ( p ) is the pth Fourier coefficient of f. Moreover, we show the existence of a positive absolute constant C0, independent of f, such that there are infinitely many pairs ( p 1 , p 2 ) of distinct primes satisfying | p 1 − p 2 | ≤ C 0 and λ f ( p 1 ) λ f ( p 2 ) ≠ 0 , which presents a modular analogue of the recent work of Maynard and Zhang on bounded gaps between primes.
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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 | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| 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