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

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 $a_1\ne a_2$ oscillates like a cosine, when renormalized by elementary functions. Moreover, we give asymptotic formulas for arbitrary linear combinations of $\{p(a,b,n){\}}_{1\le a\le b}$.

Typos and grammatical errors fixed