Large sums of symmetric power coefficients of holomorphic cusp forms
Bibliographic record
Abstract
Given a non-CM primitive cusp form f of even weight k and level N , we let s y m m f denote the m -th symmetric power lift of f . We denote by { λ s y m m f ( n ) } n ∈ N the sequence of normalized coefficients of the Dirichlet series associated to the L -function of s y m m f . In this paper, we investigate the range of x (in terms of k and N ) for which there are cancellations in the sum S ( x , s y m m f ) = ∑ n ≤ x λ s y m m f ( n ) . We first prove that S ( x , s y m 2 f ) = o ( x log 2 x ) implies that λ s y m 2 f ( n ) < 0 for some n ≤ x . Assuming the Generalized Riemann Hypothesis (GRH) for L ( s , s y m m f ) , we also show that S ( x , s y m m f ) = o m ( x log m x ) in the range log x / log log k N → ∞ and S ( x , s y m m f ) = o m , ϵ ( x ) in the range x > ( k N ) ϵ .
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How this classification was reachedexpand
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 | 0.002 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 0.003 |
| 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) | 0.000 | 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".