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

TL;DR

This paper extends previous work on the asymptotic behavior of partial fractions in the generating function for restricted partitions by providing bounds on error terms using advanced saddle-point techniques and analysis of dilogarithm zeros.

Contribution

It introduces new bounds on error terms in the partial fraction decomposition of the partition generating function, utilizing refined saddle-point analysis and properties of the dilogarithm.

Findings

Bounded the error terms in the asymptotic expansion

Developed estimates for products of sines in the context of saddle-point analysis

Linked saddle-points to zeros of the analytically continued dilogarithm

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. In part I, we studied the behavior of coefficients in the partial fraction decomposition of this product as $N \to \infty$ by applying the saddle-point method to get the asymptotics of the main terms. In this second part we bound the error terms. This involves estimating products of sines and further saddle-point arguments. The saddle-points needed are associated to zeros of the analytically continued dilogarithm.