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

2021/03/17 by Kathrin Bringmann, Min-Joo Jang, Bringmann, Kathrin +4
Mathematics · #11E45 #11F11 #11F37 #Advanced Algebra and Geometry #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2103.09653

openalex publication_date 2021/03/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 (1)/(16) times the number of such representations with arbitrary integer parameters (often called generalized polygonal numbers).

Citations

• On Modular Forms of Half Integral Weight

Related

• Fermat's polygonal number theorem for repeated generalized polygonal numbers
• Weighted sums of generalized polygonal numbers with coefficients 1 or 2
• Conjectures of Sun about sums of polygonal numbers
• Sums of four polygonal numbers: precise formulas
• Sums of generalized polygonal numbers of almost prime "length"