The Cubic Chan–Chua Conjecture

Abstract

A conjecture that expresses the {$n$}th power of the cubic theta function $a(q)=\sum_j\sum_k q^{j^2+jk+k^2}$ in terms of Eisenstein series is formulated. It is an analogue of four conjectures of H. H. Chan and K. S. Chua for powers of $\varphi^2(q)=\sum_j\sum_k q^{j^2+k^2}$. With the help of a computer, the conjecture is shown to be true for $6\leq n \leq 100$. It is conjectured that the result continues to hold for $n>100$.