Elementary Evaluation of Convolution Sums involving the Sum of Divisors Function for a Class of positive Integers

We discuss an elementary method for the evaluation of the convolution sums $\underset{\substack{ {(l,m)\in\mathbb{N}_{0}^{2}} \\ {\alpha\,l+\beta\,m=n} } }{\sum}\sigma(l)\sigma(m)$ for those $\alpha,\beta\in\mathbb{N}$ for which $\gcd{(\alpha,\beta)}=1$ and $\alpha\beta=2^{\nu}\mho$, where $\nu\in\{0,1,2,3\}$ and $\mho$ is a finite product of distinct odd primes. Modular forms are used to achieve this result. We also generalize the extraction of the convolution sum to all natural numbers. Formulae for the number of representations of a positive integer $n$ by octonary quadratic forms using convolution sums belonging to this class are then determined when $\alpha\beta\equiv 0\pmod{4}$ or $\alpha\beta\equiv 0\pmod{3}$. To achieve this application, we first discuss a method to compute all pairs $(a,b),(c,d)\in\mathbb{N}^{2}$ necessary for the determination of such formulae for the number of representations of a positive integer $n$ by octonary quadratic forms when $\alpha\beta$ has the above form and $\alpha\beta\equiv 0\pmod{4}$ or $\alpha\beta\equiv 0\pmod{3}$. We illustrate our approach by explicitly evaluating the convolution sum for $\alpha\beta=33=3\cdot 11,\> \alpha\beta=40=2^{3}\cdot 5$ and $\alpha\beta=56=2^{3}\cdot 7$, and by revisiting the evaluation of the convolution sums for $\alpha\beta=10$, $11$, $12$, $15$, $24$. We then apply these convolution sums to determine formulae for the number of representations of a positive integer $n$ by octonary quadratic forms. In addition, we determine formulae for the number of representations of a positive integer $n$ when $(a,b)=(1,1)$, $(1,3)$, $(2,3)$, $(1,9)$.