Evaluation of the convolution sum $\sum_{al+bm=n} \sigma(l) \sigma(m)$ for $(a,b)=(1,48),(3,16),(1,54),(2,27)$

Abstract

We determine the convolution sum $\sum_{al+bm=n} \sigma(l) \sigma(m)$ for $(a,b)=(1,48), (3, 16), (1,54), (2,27)$ for all positive integers $n$. We then use these evaluations together with known evaluations of other convolution sums to determine the numbers of representations of $n$ by the octonary quadratic forms $k(x_1^2 + x_1x_2 + x_2^2 +x_3^2 + x_3x_4 + x_4^2) + l(x_5^2 + x_5x_6 + x_6^2 + x_7^2 + x_7x_8 + x_8^2)$ for $(k,l)=(1,16), (1,18), (2,9)$. A modular form approach is used.