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

Abstract

Let $0<n,\alpha ,\beta \in \mathbb{N}$ be such that $\gcd (\alpha ,\beta )=1$. We carry out the evaluation of the convolution sums $\sum _{\begin{array}{c}(k,l)\in {\mathbb{N}}^2\\ \alpha k+\beta l=n\end{array}}\sigma (k){\sigma }_3(l)$ and $\sum _{\begin{array}{c}(k,l)\in {\mathbb{N}}^2\\ \alpha k+\beta l=n\end{array}}{\sigma }_3(k)\sigma (l)$ 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 $\underset{i=1}{\sum ^{12}}{x}_i^2$ when the level $\alpha \beta \equiv 0(\mathrm{mod}4)$, and $\underset{i=1}{\sum ^6}(x_{2i-1}^2+x_{2i-1}{x}_{2i}+x_{2i}^2)$ when the level $\alpha \beta \equiv 0(\mathrm{mod}3)$. Our approach is then illustrated by explicitly evaluating the convolution sum for $\alpha \beta =3$, $4$, $6$, $7$, $8$, $9$, $12$, $14$, $15$, $16$, $18$, $20$, $21$, $27$, $32$. 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.

51 pages, 9 tables. arXiv admin note: text overlap with arXiv:1609.01343, arXiv:1607.01082