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arXiv · 2111.04183

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

Walter Bridges·Johann Franke·Taylor Garnowski
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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 $\{p(a,b,n)\}_{1 \leq a \leq b}$.

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Walter Bridges, Johann Franke, Taylor Garnowski. 2021-11-07. Asymptotics for the twisted eta-product and applications to sign changes in partitions. https://arxiv.org/abs/2111.04183

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