New properties of multiple harmonic sums modulo đ and đ-analogues of Leshchinerâs series
Bibliographic record
Abstract
In this paper we present some new binomial identities for multiple harmonic sums whose indices are the sequences ( { 1 } a , c , { 1 } b ) , (\{1\}^a,c,\{1\}^b), ( { 2 } a , c , { 2 } b ) (\{2\}^a,c,\{2\}^b) and prove a number of congruences for these sums modulo a prime p . p. The congruences obtained allow us to find nice p p -analogues of Leshchinerâs series for zeta values and to refine a result due to M. Hoffman and J. Zhao about the set of generators of the multiple harmonic sums of weight 7 7 and 9 9 modulo p p . As a further application we provide a new proof of Zagierâs formula for ζ â ( { 2 } a , 3 , { 2 } b ) \zeta ^{*}(\{2\}^a,3,\{2\}^b) based on a finite identity for partial sums of the zeta-star series.
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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 | 0.004 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.002 | 0.002 |
| Scholarly communication | 0.003 | 0.004 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.014 | 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 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".