Generalized polygonal number representations

Abstract

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 $r_n^{\square }(8(k-2)N+n(k-4{)}^2)$ for any $k\ge 3$ and any $n\ge 4$. Consequently, we determine the asymptotics of ${\sum }_{N\le x}{r}_4^k(N{)}^2$ and, via a result of Bringmann, Jang, Kane, and Tse, prove a similar closed-form asymptotic relation between the number $r_{4,+}^k(N)$ of representations of $N$ as the sum of four ordinary $k$-gonal numbers and $r_4^{\square }(8(k-2)N+n(k-4{)}^2)$. We also show that if $4\mid k$, any strictly increasing infinite subsequence on which $r_{4,+}^k$ is bounded converges $2$-adically to $(k-4{)}^2/(4-2k)\in {\mathbb{Z}}_2$, supplementing a result of Meng and Sun, and if $4\nmid k$, there is no strictly increasing infinite subsequence on which $r_{4,+}^k$ is bounded.

41 pages