Disproof of a conjecture by Rademacher on partial fractions (Q2814367)

The Rademacher coefficients \(C_{h,k,l}(N)\) arise in the ordinary partial fraction decomposition of the generating function for partitions of integers into at most \(N\) parts NEWLINE\[NEWLINE\frac{1}{(x;x)_N} = \sum_{k=1}^{N} \sum_{0\leqslant h<k \atop \gcd(h,k)=1 } \sum_{l=1}^{\lfloor N/k \rfloor} \frac{C_{h,k,l}(N)}{(x-e^{2\pi i h/k})^l}.NEWLINE\]NEWLINE For all integer \(h,k,l\) such that \(0\leqslant h< k\), \(\gcd(h,k)=1\) and \(l\geqslant 1\), \textit{H. Rademacher} conjectured in his classical book [Topics in analytic number theory. York: Springer-Verlag (1973; Zbl 0253.10002)] that the limit \(\lim_{N\to\infty} C_{h,k,l}(N)\) exists and equals a particular value that he specified. In this paper, the authors disproved this conjecture and confirm \textit{A. V. Sills} and \textit{D. Zeilberger}'s observations [J. Difference Equ. Appl. 19, No. 4, 680--689 (2013; Zbl 1301.11069)].