Generalized polygonal number representations

Abstract: For $k\geq5$ and $n\geq 4$, let $r_n^{(k)}(N)$ be the number of representations of $N$ as the sum of $n$ generalized $k$-gonal numbers and $r_n^{\square}(N)$ be the number of representations of $N$ as the sum of $n$ squares. By modifying the Heath-Brown circle method, we prove a closed-form asymptotic relation between $r_{n}^{(k)}(N)$ and $r_n^{\square}(N)$ for $k\not\equiv 0\bmod 4$ and any $n\geq4$. Consequently, we relate the number of representations of $N$ as the sum of four ordinary $k$-gonal numbers to $r_4^{\square}(N)$ via a result of Bringmann--Jang--Kane--Tse.

number theory

Audience: researchers in the topic

Combinatorial and additive number theory seminar (CANT 2026)