A continued fraction of Ramanujan and some Ramanujan-Weber class invariants
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Bibliographic record
Abstract
On Page 36 of his “lost” notebook, Ramanujan recorded four q-series representations of the famous Rogers-Ramanujan continued fraction. In this paper, we establish two q-series representations of Ramanujan’s continued fraction found in his “lost” notebook. We also establish three equivalent integral representations and modular equations for a special case of this continued fraction. Furthermore, we derive continued-fraction representations for the Ramanujan-Weber class invariants 1n and Gn and establish formulas connecting 1n and Gn. We obtain relations between our continued fraction with the Ramanujan-Göllnitz-Gordon and Ramanujan’s cubic continued fractions. Finally, we find some algebraic numbers and transcendental numbers associated with a certain continued fraction A(q) which is related to Ramanujan’s continued fraction F(a,b,λ,q), the Ramanujan-Göllnitz-Gordon continued fraction H(q) and the Dedekind eta function η(s).
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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 | 0.002 |
| 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 | 0.001 |
| 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)
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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