Rademacher's infinite partial fraction conjecture is (almost certainly) false

Andrew V. Sills, Doron Zeilberger

Research output: Contribution to journalArticlepeer-review

4 Scopus citations

Abstract

In his book Topics in Analytic Number Theory, Hans Rademacher conjectured that the limits of certain sequences of coefficients that arise in the ordinary partial fraction decomposition of the generating function for partitions of integers into at most N parts exist and equal particular values that he specified. Despite being open for nearly four decades, little progress has been made towards proving or disproving the conjecture, perhaps in part due to the difficulty in actually computing the coefficients in question. In this paper, we present a recurrence (alias difference equation) which provides a fast algorithm for calculating the Rademacher coefficients, a large amount of data, direct formulae for certain collections of Rademacher coefficients, and overwhelming evidence against the truth of the conjecture. While the limits of the sequences of Rademacher coefficients do not exist (the sequences oscillate and attain arbitrarily large positive and negative values), the sequences do get very close to Rademacher's conjectured limits for certain (predictable) indices in the sequences.

Original languageEnglish (US)
Pages (from-to)680-689
Number of pages10
JournalJournal of Difference Equations and Applications
Volume19
Issue number4
DOIs
StatePublished - Apr 2013

All Science Journal Classification (ASJC) codes

  • Analysis
  • Algebra and Number Theory
  • Applied Mathematics

Keywords

  • Rademacher conjecture
  • partial fractions
  • partitions
  • recurrence

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