#### Document Type

Article

#### Publication Date

2013

#### Publication Title

Journal of Difference Equations and Applications

#### DOI

10.1080/10236198.2012.678837

#### ISSN

1563-5120

#### 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.

#### Recommended Citation

Sills, Andrew, Doron Zeilberger.
2013.
"Rademacher's Infinite Partial Fractions Conjecture is (Almost Certainly) False."
*Journal of Difference Equations and Applications*, 19 (4): 680-689.
doi: 10.1080/10236198.2012.678837 source: http://arxiv.org/abs/1110.4932

https://digitalcommons.georgiasouthern.edu/math-sci-facpubs/181

## Comments

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Creative Commons Attribution license,Creative Commons Attribution-Noncommercial-ShareAlike license, orCreate Commons Public Domain Declaration. The publisher's final edited version of this article is available atJournal of Diﬀerence Equations and Application.