Ramanujan Summation and the Exponential Generating Function $ \sum_{k=0}^{\infty}\frac{z^{k}}{k!}\zeta ^{\prime}(-k)$
Bibliographic record
Abstract
In the sixth chapter of his notebooks Ramanujan introduced a method of summing divergent series which assigns to the series the value of the associated Euler-MacLaurin constant that arises by applying the Euler-MacLaurin summation formula to the partial sums of the series. This method is now called the Ramanujan summation process. In this paper we calculate the Ramanujan sum of the exponential generating functions $\sum_{n\geq 1}\log n e^{nz}$ and $\sum_{n\geq 1}H_n^{(j)} e^{-nz}$ where $H_n^{(j)}=\sum_{m=1}^n \frac{1}{m^j}$. We find a surprising relation between the two sums when $j=1$ from which follows a formula that connects the derivatives of the Riemann zeta - function at the negative integers to the Ramanujan summation of the divergent Euler sums $\sum_{n\ge 1} n^kH_n, k \ge 0$, where $H_n= H_n^{(1)}$. Further, we express our results on the Ramanujan summation in terms of the classical summation process called the Borel sum.
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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 | 0.001 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.002 | 0.001 |
| Research integrity | 0.001 | 0.002 |
| 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".