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

TL;DR

This paper presents a new algorithm and extensive data analysis that strongly suggest Rademacher's conjecture about the limits of certain coefficients in partition generating functions is false, showing the sequences oscillate wildly.

Contribution

The authors provide a fast algorithm, extensive computational data, and formulas that collectively disprove Rademacher's conjecture about the limits of specific coefficients.

Findings

Sequences oscillate and do not have limits.

Sequences attain arbitrarily large positive and negative values.

Sequences sometimes approximate the conjectured limits before diverging.

Abstract

In his book \emph{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 toward proving or disproving the conjecture, perhaps in part due to the difficulty in actually computing the coefficients in question. In this paper, we provide a fast algorithm for calculating the Rademacher coefficients, a large amount of data, direct formulas 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.