Poly-Bernoulli numbers and related zeta functions
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Bibliographic record
Abstract
In this expository article, we review some aspects of poly-Bernoulli numbers and related zeta functions. The poly-Bernoulli number is a generalization of the classical Bernoulli number using the polylogarithm series. Although its definition looks rather artificial at first glance, it has turned out recently that the poly-Bernoulli numbers of negative index have very nice combinatorial interpretations, and also they appear in special values of certain zeta functions. It may therefore be reasonable to seek arithmetic properties that may be involved with poly-Bernoulli numbers. The author made one such attempt with late Arakawa in the hope of finding a nice zeta function which connects poly-Bernoulli numbers with the so-called multiple zeta values, the subject of wide interest not only in number theory but also in numerous other branches such as topology, quantum groups, arithmetic geometry, mathematical physics etc. This work with Arakawa will be reviewed in §3, after recalling definitions and properties of poly-Bernoulli numbers in §2. In §4 we give some results and speculations concerning the “multiple harmonic sums mod p” and “multiple zeta-star values.” In the final section, §5, we discuss a different type of zeta function which also has some relation to poly-Bernoulli numbers as well as to certain generalized multiple zeta values. The author would like to take this opportunity to express his deep gratitude to late Professor Tsuneo Arakawa on the occasion of his sixtieth birthday, whose encouragement and interest at the early stage of the research on this topic greatly helped in developing the work further.
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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 | 0.001 | 0.001 |
| Science and technology studies | 0.002 | 0.002 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| 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