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A third-order Apery-like recursion for $\zeta(5)$

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Wadim Zudilin (Moscow)

Number Theory · Submitted 18 Jun 2002

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim The note contains a possible generalization of the above results for the number $\zeta(5)$.

how it is stated · hedged possibility 0.99 read the record →

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arXiv IDmath/0206178 (version 2)
Pith ID5ZG7TLJC, Pith's permanent record of this paper and its review
CitationWadim Zudilin (Moscow). "A third-order Apery-like recursion for $\zeta(5)$." arXiv:math/0206178 (2002). https://pith.science/paper/5ZG7TLJC
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The compiled record

The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,597 input tokens.

What the abstract itself states, as probabilities: claims first 0.09 · states limitation 0.57 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.09 · compares with prior results 0.89 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; .

In 1978, Apery has given sequences of rational approximations to $\zeta(2)$ and $\zeta(3)$ yielding the irrationality of each of these numbers. One of the key ingredient of Apery's proof are second-order difference equations with polynomial coefficients satisfied by numerators and denominators of the above approximations. Recently, a similar second-order difference equation for $\zeta(4)$ has been discovered. The note contains a possible generalization of the above results for the number $\zeta(5)$.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 In 1978, Apery has given sequences of rational approximations to $\zeta(2)$ and $\zeta(3)$ yielding the irrationality of each of these numbers. background 1.00 0.03 0.04 asserted 1.00
s1 One of the key ingredient of Apery's proof are second-order difference equations with polynomial coefficients satisfied by numerators and denominators of the above approximations. background 0.99 0.04 0.03 asserted 1.00
s2 Recently, a similar second-order difference equation for $\zeta(4)$ has been discovered. background 0.98 0.10 0.05 asserted 1.00
s3 The note contains a possible generalization of the above results for the number $\zeta(5)$. contribution 0.98 0.83 0.46 hedged possibility 0.99

What kind of paper the classifier reports this is: theoretical result 0.96 · evidence from proof or derivation 0.91.

What was examined, and what was not

sources read
  • arXiv listing abstract v2
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

In 1978, Apery has given sequences of rational approximations to $\zeta(2)$ and $\zeta(3)$ yielding the irrationality of each of these numbers. One of the key ingredient of Apery's proof are second-order difference equations with polynomial coefficients satisfied by numerators and denominators of the above approximations. Recently, a similar second-order difference equation for $\zeta(4)$ has been discovered. The note contains a possible generalization of the above results for the number $\zeta(5)$.

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Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

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