Polynomial partition asymptotics

Abstract

Let $f\in \mathbb{Z}[y]$ be a polynomial such that $f(\mathbb{N})\subseteq \mathbb{N}$, and let $p_{{\mathcal{A}}_f}(n)$ denote number of partitions of $n$ whose parts lie in the set ${\mathcal{A}}_f:=\{f(n):n\in \mathbb{N}\}$. Under hypotheses on the roots of $f-f(0)$, we use the Hardy--Littlewood circle method, a polylogarithm identity, and the Matsumoto--Weng zeta function to derive asymptotic formulae for $p_{{\mathcal{A}}_f}(n)$ as $n$ tends to infinity. This generalises asymptotic formulae for the number of partitions into perfect $d$th powers, established by Vaughan for $d=2$, and Gafni for the case $d\ge 2$, in 2015 and 2016 respectively.

arXiv comment26 pages, Improved exposition throughout and several typos fixed