TY - JOUR AU - Ntienjem, Ebenezer PY - 2021/02/04 Y2 - 2024/03/29 TI - Evaluation of Convolution Sums entailing mixed Divisor Functions for a Class of Levels JF - New Zealand Journal of Mathematics JA - NZ J Math VL - 50 IS - 0 SE - Articles DO - 10.53733/80 UR - https://www.nzjmath.org/index.php/NZJMATH/article/view/80 SP - 125--180 AB - <p>Let $0&lt; n,\alpha,\beta\in\mathbb{N}$ be such that $\gcd{(\alpha,\beta)}=1$. We carry out the evaluation of the convolution sums $\underset{\substack{ {(k,l)\in\mathbb{N}^{2}} \\ {\alpha\,k+\beta\,l=n} } }{\sum}\sigma(k)\sigma_{3}(l)$ and $\underset{\substack{ {(k,l)\in\mathbb{N}^{2}} \\ {\alpha\,k+\beta\,l=n} } }{\sum}\sigma_{3}(k)\sigma(l)$ for all levels $\alpha\beta\in\mathbb{N}$, 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 $\alpha\beta\equiv 0\pmod{4}$, and $\underset{i=1}{\overset{6}{\sum}}\,(\,x_{2i-1}^{2}+ x_{2i-1}x_{2i} + x_{2i}^{2}\,)$ when the level $\alpha\beta\equiv 0\pmod{3}$. Our approach is then illustrated by explicitly evaluating the convolution sum for $\alpha\beta=3$, $6$, $7$, $8$, $9$, $12$, $14$, $15$, $16$, $18$, $20$, $21$, $27$, $32$. 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.</p> ER -