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Disproof of a conjecture by Rademacher on partial fractions

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Michael Drmota, Stefan Gerhold

Number Theory · Submitted 16 Dec 2013

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

The authors' main claim We carry out an asymptotic analysis that disproves this conjecture, thus confirming recent observations of Sills and Zeilberger (Journal of Difference Equations and Applications 19, 2013), who gave strong numerical evidence against the conjecture.

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arXiv ID1312.4289 (version 1)
Pith IDRDF6RN6S, Pith's permanent record of this paper and its review
CitationMichael Drmota, Stefan Gerhold. "Disproof of a conjecture by Rademacher on partial fractions." arXiv:1312.4289 (2013). https://pith.science/paper/RDF6RN6S
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Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,388 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.07 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.07 · compares with prior results 0.93 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

In his book Topics in Analytic Number Theory, Rademacher considered the generating function of partitions into at most $N$ parts, and conjectured certain limits for the coefficients of its partial fraction decomposition. We carry out an asymptotic analysis that disproves this conjecture, thus confirming recent observations of Sills and Zeilberger (Journal of Difference Equations and Applications 19, 2013), who gave strong numerical evidence against the conjecture.

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#sentencerolethe authors' own workstates the main conclusionstrength
s0 In his book Topics in Analytic Number Theory, Rademacher considered the generating function of partitions into at most $N$ parts, and conjectured certain limits for the coefficients of its partial fraction decomposition. background 1.00 0.03 0.03 asserted 0.90
s1 We carry out an asymptotic analysis that disproves this conjecture, thus confirming recent observations of Sills and Zeilberger (Journal of Difference Equations and Applications 19, 2013), who gave strong numerical evidence against the conjecture. result 0.78 0.92 0.96 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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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 his book Topics in Analytic Number Theory, Rademacher considered the generating function of partitions into at most $N$ parts, and conjectured certain limits for the coefficients of its partial fraction decomposition. We carry out an asymptotic analysis that disproves this conjecture, thus confirming recent observations of Sills and Zeilberger (Journal of Difference Equations and Applications 19, 2013), who gave strong numerical evidence against the conjecture.

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