Evaluation of convolution sums $$\sum \limits _{l+15m=n} \sigma (l) \sigma (m)$$ ∑ l + 15 m = n σ ( l ) σ ( m ) and $$\sum \limits _{3l+5m=n} \sigma (l) \sigma (m)$$ ∑ 3 l + 5 m = n σ ( l ) σ ( m )

In this article, we have evaluated the convolution sums $$\displaystyle \sum _{al+bm=n} \sigma (l) \sigma (m)$$ ∑ a l + b m = n σ ( l ) σ ( m ) for $$ a\cdot b =15$$ a · b = 15 , where $$a,b \in \mathbb {N}$$ a , b ∈ N , using an elementary method. The deduced convolution sums are in a little elegent form than that derived by B. Ramakrishnan and B. Sahu [24]. As a consequence, we determine a formula for the number of representations of a positive integer n by the octonary quadratic form $$\begin{aligned} \left( x_1^2+x_1x_2+x_2^2+x_3^2+x_3x_4+x_4^2\right) +5\left( x_5^2+x_5x_6+x_6^2+x_7^2+x_7x_8+x_8^2\right) . \end{aligned}$$ x 1 2 + x 1 x 2 + x 2 2 + x 3 2 + x 3 x 4 + x 4 2 + 5 x 5 2 + x 5 x 6 + x 6 2 + x 7 2 + x 7 x 8 + x 8 2 .