Zeros of classical Eisenstein series and recent developments
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Bibliographic record
Abstract
Abstract. In this survey, we begin by recalling a beautiful result of F. K. C. Rankin and Swinnerton-Dyer on the location of zeros of the classical Eisenstein series Ek(z) for the full modular group. We then explore more recent studies which have built upon this work to analyze the behavior of zeros of Ek(z), such as work by Nozaki on their separation property. We also review similar results for other classes of modular forms as well as zeros of Eisenstein series for different groups. We conclude with some results of the authors for a family of odd weight Eisenstein series on Γ(2), a prototypical genus zero subgroup with a simple fundamental domain. 1 Eisenstein series on SL2(Z) We begin this survey by defining modular forms on the full modular group Γ:=SL2(Z). Let H denote the complex upper half-plane
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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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 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) | 0.006 | 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