Asymptotics for the partial fractions of the restricted partition generating function I

TL;DR

This paper analyzes the asymptotic behavior of partial fraction coefficients of the restricted partition generating function as the number of parts grows large, using saddle-point methods linked to dilogarithm zeros, and refutes a previous conjecture.

Contribution

It provides the first detailed asymptotic analysis of these coefficients and disproves Rademacher's conjecture using advanced complex analysis techniques.

Findings

Asymptotic formulas for partial fraction coefficients as N approaches infinity

Identification of the saddle-point related to dilogarithm zeros

Disproof of Rademacher's conjecture on partition generating functions

Abstract

The generating function for $p_N(n)$, the number of partitions of $n$ into at most $N$ parts, may be written as a product of $N$ factors. We find the behavior of coefficients in the partial fraction decomposition of this product as $N \to \infty$ by applying the saddle-point method, where the saddle-point we need is associated to a zero of the analytically continued dilogarithm. Our main result disproves a conjecture of Rademacher.