The H and K Family of Mock Theta Functions
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Bibliographic record
Abstract
Abstract In his last letter to Hardy, Ramanujan defined 17 functions F ( q ), | q | < 1, which he calledmock θ -functions. He observed that as q radially approaches any root of unity ζ at which F ( q ) has an exponential singularity, there is a θ -function T ζ ( q ) with F ( q ) − T ζ ( q ) = O (1). Since then, other functions have been found that possess this property. These functions are related to a function H ( x , q ), where x is usually q r or e 2π ir for some rational number r . For this reason we refer to H as a “universal” mock θ -function. Modular transformations of H give rise to the functions K , K 1 , K 2. The functions K and K 1 appear in Ramanujan's lost notebook. We prove various linear relations between these functions using Appell–Lerch sums (also called generalized Lambert series). Some relations (mock theta “conjectures”) involving mock θ -functions of even order and H are listed.
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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 | 0.002 |
| 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 | 0.000 |
| 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 it