Asymptotics for the twisted eta-product and applications to sign changes in partitions

TL;DR

This paper derives asymptotic formulas for coefficients of twisted eta-products and applies them to analyze sign changes and oscillations in partition functions with congruence conditions.

Contribution

It provides new asymptotic formulas for twisted eta-products and applies these to understand sign changes and oscillations in partition counts with residue class restrictions.

Findings

Difference in partition counts oscillates like a cosine as n grows large.

Asymptotic formulas include secondary terms for partition functions with congruence conditions.

Results apply to arbitrary linear combinations of partition functions.

Abstract

We prove asymptotic formulas for the complex coefficients of $(\zeta q;q)_\infty^{-1}$, where $\zeta$ is a root of unity, and apply our results to determine secondary terms in the asymptotics for $p(a,b,n)$, the number of integer partitions of $n$ with largest part congruent $a$ modulo $b$. Our results imply that, as $n \to \infty$, the difference $p(a_1,b,n)-p(a_2,b,n)$ for $a_1 \neq a_2$ oscillates like a cosine, when renormalized by elementary functions. Moreover, we give asymptotic formulas for arbitrary linear combinations of $\{p(a,b,n)\}_{1 \leq a \leq b}$.