Some Convolution Sums and The Representation Numbers

Abstract

We evaluate the convolution sums

\(\sum_{l+30m=n} \sigma(l) \sigma(m), \sum_{3l+10m=n} \sigma(l) \sigma(m),\\ \)
\(\sum_{2l+15m=m} \sigma(l) \sigma(m), \sum_{5l+6m=n} \sigma(l) \sigma(m), \\\)
\(\sum_{l+33m=n} \sigma(l) \sigma(m), \sum_{3l+11m=n} \sigma(l) \sigma(m), \\\)
\(\sum_{l+39m=n} \sigma(l) \sigma(m), \sum_{3l+13m=n} \sigma(l) \sigma(m),\\\)

for all \(n \in \mathbb{N}\) using the theory of quasimodular forms, and utilize these convolution sums to determine the number of representations of a positive integer \(n\) by the forms
\[x_1^2 +x_1x_2+ x_2^2 + x_3^2 +x_3x_4+ x_4^2
+ a(x_5^2 + x_5x_6+x_6^2 + x_7^2 + x_7x_8+x_8^2), \]
for \(a = 10, 11, 13\). Quasimodular forms, divisor functions, convolution sums, representation number \(11A25,11F11,11F25,11F20\)