Evaluation of Convolution Sums entailing mixed Divisor Functions for a Class of Levels

Let $0< n,α,β\in\mathbb{N}$ be such that $\gcd{(α,β)}=1$. We carry out the evaluation of the convolution sums $\underset{\substack{ {(k,l)\in\mathbb{N}^{2}} \\ {α\,k+β\,l=n} } }{\sum}σ(k)σ_{3}(l)$ and $\underset{\substack{ {(k,l)\in\mathbb{N}^{2}} \\ {α\,k+β\,l=n} } }{\sum}σ_{3}(k)σ(l)$ for all levels $αβ\in\mathbb{N}$, by using in particular modular forms. We next apply convolution sums belonging to this class of levels to determine formulae for the number of representations of a positive integer $n$ by the quadratic forms in twelve variables $\underset{i=1}{\overset{12}{\sum}}x_{i}^{2}$ when the level $αβ\equiv 0\pmod{4}$, and $\underset{i=1}{\overset{6}{\sum}}\,(\,x_{2i-1}^{2}+ x_{2i-1}x_{2i} + x_{2i}^{2}\,)$ when the level $αβ\equiv 0\pmod{3}$. Our approach is then illustrated by explicitly evaluating the convolution sum for $αβ=3$, $4$, $6$, $7$, $8$, $9$, $12$, $14$, $15$, $16$, $18$, $20$, $21$, $27$, $32$. These convolution sums are then applied to determine explicit formulae for the number of representations of a positive integer $n$ by quadratic forms in twelve variables.

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