Dirichlet Functions Generated by Blaschke Products
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Bibliographic record
Abstract
The continuation of general Dirichlet series to meromorphic functions in the complex plane remains an outstanding problem. It has been completely solved only for Dirichlet L-series. A sufficient condition for the general case exists, however it is impossible to verify that it is fulfilled. We solve this problem here for another class of general Dirichlet series, namely those series which are obtained from infinite Blaschke products by a particular change of variable. This is a source of examples of general Dirichlet series with infinitely many poles. An interesting new case is now revealed, in which the singular points of the extended function form a continuum. We take a closer look at the case of Dirichlet series with natural boundary and give examples of such series. Some figures illustrate the theory.
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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) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.003 | 0.002 |
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