Sums of triangular numbers and sums of squares
Abstract
For nonnegative 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 BatemanDatskovskyKnopp'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.
 Publication:

arXiv eprints
 Pub Date:
 July 2021
 arXiv:
 arXiv:2107.00787
 Bibcode:
 2021arXiv210700787A
 Keywords:

 Mathematics  Number Theory;
 11F11;
 11E25;
 11F27