Sums of triangular numbers and sums of squares

TL;DR

This paper investigates relationships between sums of triangular numbers and sums of squares with coefficients 1 or 3, extending known identities to larger parameters and establishing asymptotic equivalences using Eisenstein series analysis.

Contribution

It extends known identities for representation numbers to cases where a+3b>8 and proves asymptotic equivalences for general a, b with a+b even, using Eisenstein series instead of the circle method.

Findings

Established asymptotic equivalence of representation formulas for large n.

Extended identities to cases with a+3b>8.

Developed a robust method based on Eisenstein series analysis.

Abstract

For non-negative integers $a,b,$ and $n$, let $N(a, b; n)$ be the number of representations of $n$ as a sum of squares with coefficients $1$ or $3$ ($a$ of ones and $b$ of threes). Let $N^*(a,b; n)$ be the number of representations of $n$ as a sum of odd squares with coefficients $1$ or $3$ ($a$ of ones and $b$ of threes). We have that $N^*(a,b;8n+a+3b)$ is the number of representations of $n$ as a sum of triangular numbers with coefficients $1$ or $3$ ($a$ of ones and $b$ of threes). It is known that for $a$ and $b$ satisfying $1\leq a+3b \leq 7$, we have $$ N^*(a,b;8n+a+3b)= \frac{2}{2+{a\choose4}+ab} N(a,b;8n+a+3b) $$ and for $a$ and $b$ satisfying $a+3b=8$, we have $$ N^*(a,b;8n+a+3b) = \frac{2}{2+{a\choose4}+ab} \left( N(a,b;8n+a+3b) - N(a,b; (8n+a+3b)/4) \right). %& t(8,0;{n}) = \frac{1}{36} \left( N(8,0;8n+8) - N(8,0;2n+2) \right). \label{eq31_5} $$ Such identities are not known for $a+3b>8$. In this paper, for general $a$ and $b$ with $a+b$ even, we prove asymptotic equivalence of formulas similar to the above, as $n\rightarrow\infty$. One of our main results extends a theorem of Bateman, Datskovsky, and Knopp where the case $b=0$ and general $a$ was considered. Our approach is different from Bateman-Datskovsky-Knopp's proof where the circle method and singular series were used. We achieve our results by explicitly computing the Eisenstein components of the generating functions of $N^*(a,b;8n+a+3b)$ and $N(a,b;8n+a+3b)$. The method we use is robust and can be adapted in studying the asymptotics of other representation numbers with general coefficients.