SOME FAMILIES OF RAPIDLY CONVERGENT SERIES REPRESENTATIONS FOR THE ZETA FUNCTIONS
Bibliographic record
Abstract
Many interesting families of rapidly convergent series representations for the Riemann Zeta function $\zeta (2n+1)$ $(n\in {\Bbb N})$ were considered recently by various authors. In this survey-cum-expository paper, the author presents a systematic (and historical) investigation of these series representations. Relevant connections of the results presented here with several other known series representations for $\zeta (2n+1)$ $(n\in {\Bbb N})$ are also pointed out. In one of many computationally useful special cases presented here, it is observed that $\zeta (3)$ can be represented by means of a series which converges much faster than that in Euler's celebrated formula as well as the series used recently by Ap\'{e}ry in his proof of the irrationality of $\zeta (3)$. Symbolic and numerical computations using Mathematica (Version 4.0) for Linux show, among other things, that only 50 terms of this series are capable of producing an accuracy of seven decimal places.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.008 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.005 | 0.001 |
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 source (direct Gemma or distilled Codex), 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".