Evaluation of the Convolution Sums $\sum _{\begin{array}{c}(l,m)\in {\mathbb{N}}_0^2\alpha l+\beta m=n\end{array}}\sigma (l)\sigma (m)$, where $\alpha \beta =44,52$

Abstract

The convolution sum, $\sum _{\begin{array}{c}(l,m)\in {\mathbb{N}}_0^2\alpha l+\beta m=n\end{array}}\sigma (l)\sigma (m)$, where $\alpha \beta =44,52$, is evaluated for all natural numbers $n$. We then use these convolution sums to determine formulae for the number of representations of a natural number by the octonary quadratic forms $a(x_1^2+x_2^2+x_3^2+x_4^2)+b(x_5^2+x_6^2+x_7^2+x_8^2)$, where $(a,b)=(1,11),(1,13)$.

14 pages, 4 tables