Powers of certain theta products
Undergraduate project, Sun Yat-sen University, 2019
Z. Cheng, J. Wang and D. Ye, Powers of certain theta products, Mathematical Reports, 25 (2023), 577-587.
Project description
For \(N\in\{14,15\}\), define for \(n,k\geq1\),
\[r(n,k;N)= \begin{cases} \displaystyle\left|\left\{(x_{i})\in\mathbb{Z}^{2k}:\sum_{j=1}^{k}(x_{2j-1}^{2}+x_{2j-1}x_{2j}+2x_{2j}^{2})+2\sum_{j=k+1}^{2k}\left(x_{2j-1}^{2} +x_{2j-1}x_{2j}+2x_{2j}^{2}\right)\right\}\right|&\mbox{if $N=14$,}\\\\ \displaystyle\left|\left\{(x_{i})\in\mathbb{Z}^{2k}:\sum_{j=1}^{k}(x_{2j-1}^{2}+x_{2j-1}x_{2j}+x_{2j}^{2}) +5\sum_{j=k+1}^{2k}\left(x_{2j-1}^{2}+x_{2j-1}x_{2j}+x_{2j}^{2}\right)\right\}\right|&\mbox{if $N=15$.} \end{cases}\]In this project, we established Ramanujan-Mordell type formulas for \(r(n,k;N)\). For example, we proved that
\[r(n,k;14)\sim -\frac{4k}{B_{2k}}\frac{(-1)^{k}\sigma_{2k-1}(n)+\left(-2\right)^{k}\sigma_{2k-1}(n/2) +7^{k}\sigma_{2k-1}(n/7)+14^{k}\sigma_{2k-1}(n/14)}{(-1)^{k}+\left(-2\right)^{k}+7^{k}+14^{k}}\]as \(n\to\infty\), where \(B_{k}\) denotes the Bernoulli number, and \(\sigma_{s}(n)=\sum_{d\mid n}d^{s}\).