# Polynomial partition asymptotics

## TL;DR

This paper derives asymptotic formulas for the number of integer partitions with parts in polynomial-generated sets, generalizing previous results for perfect powers using advanced analytic number theory techniques.

## Contribution

It introduces a general method to obtain asymptotics for partitions into polynomial-generated sets, extending known results for perfect powers.

## Key findings

- Derived asymptotic formulas for partition counts in polynomial-generated sets
- Extended previous results from perfect powers to more general polynomial sets
- Utilized Hardy-Littlewood circle method and zeta functions in novel way

## 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 $d \geq 2$, in 2015 and 2016 respectively.