Representations of integers as sums of four polygonal numbers and partial theta functions

TL;DR

This paper investigates how integers can be expressed as sums of up to four polygonal numbers, revealing asymptotic equivalences in the count of such representations under certain conditions.

Contribution

It establishes an asymptotic equivalence between counts of representations with non-negative and arbitrary integer parameters for sums of polygonal numbers.

Findings

Asymptotic equivalence of representation counts

Counts with non-negative parameters are approximately 1/16 of those with arbitrary integers

Results connect polygonal number representations with partial theta functions

Abstract

In this paper, we consider representations of integers as sums of at most four distinct $m$-gonal numbers (allowing a fixed number of repeats of each polygonal number occurring in the sum). We show that the number of such representations with non-negative parameters (hence counting the number of points in a regular $m$-gon) is asymptotically the same as $\frac{1}{16}$ times the number of such representations with arbitrary integer parameters (often called generalized polygonal numbers).