New properties of multiple harmonic sums modulo đ and đ-analogues of Leshchinerâs series
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Bibliographic record
Abstract
In this paper we present some new binomial identities for multiple harmonic sums whose indices are the sequences <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis left-brace 1 right-brace Superscript a Baseline comma c comma left-brace 1 right-brace Superscript b Baseline right-parenthesis comma"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mn>1</mml:mn> <mml:msup> <mml:mo fence="false" stretchy="false">}</mml:mo> <mml:mi>a</mml:mi> </mml:msup> <mml:mo>,</mml:mo> <mml:mi>c</mml:mi> <mml:mo>,</mml:mo> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mn>1</mml:mn> <mml:msup> <mml:mo fence="false" stretchy="false">}</mml:mo> <mml:mi>b</mml:mi> </mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(\{1\}^a,c,\{1\}^b),</mml:annotation> </mml:semantics> </mml:math> </inline-formula> <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis left-brace 2 right-brace Superscript a Baseline comma c comma left-brace 2 right-brace Superscript b Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mn>2</mml:mn> <mml:msup> <mml:mo fence="false" stretchy="false">}</mml:mo> <mml:mi>a</mml:mi> </mml:msup> <mml:mo>,</mml:mo> <mml:mi>c</mml:mi> <mml:mo>,</mml:mo> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mn>2</mml:mn> <mml:msup> <mml:mo fence="false" stretchy="false">}</mml:mo> <mml:mi>b</mml:mi> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(\{2\}^a,c,\{2\}^b)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and prove a number of congruences for these sums modulo a prime <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p period"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>.</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">p.</mml:annotation> </mml:semantics> </mml:math> </inline-formula> The congruences obtained allow us to find nice <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="7"> <mml:semantics> <mml:mn>7</mml:mn> <mml:annotation encoding="application/x-tex">7</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="9"> <mml:semantics> <mml:mn>9</mml:mn> <mml:annotation encoding="application/x-tex">9</mml:annotation> </mml:semantics> </mml:math> </inline-formula> modulo <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . As a further application we provide a new proof of Zagierâs formula for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="zeta Superscript asterisk Baseline left-parenthesis left-brace 2 right-brace Superscript a Baseline comma 3 comma left-brace 2 right-brace Superscript b Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi> ζ </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo> â </mml:mo> </mml:mrow> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mn>2</mml:mn> <mml:msup> <mml:mo fence="false" stretchy="false">}</mml:mo> <mml:mi>a</mml:mi> </mml:msup> <mml:mo>,</mml:mo> <mml:mn>3</mml:mn> <mml:mo>,</mml:mo> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mn>2</mml:mn> <mml:msup> <mml:mo fence="false" stretchy="false">}</mml:mo> <mml:mi>b</mml:mi> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\zeta ^{*}(\{2\}^a,3,\{2\}^b)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> based on a finite identity for partial sums of the zeta-star series.
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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 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.004 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| 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