On the representations of a positive integer by certain classes of quadratic forms in eight variables

Abstract

In this paper we use the theory of modular forms to find formulas for the number of representations of a positive integer by certain class of quadratic forms in eight variables, viz., forms of the form $a_1x_1^2 + a_2 x_2^2 + a_3 x_3^2 + a_4 x_4^2 + b_1(x_5^2+x_5x_6 + x_6^2) + b_2(x_7^2+x_7x_8 + x_8^2)$, where $a_1\le a_2\le a_3\le a_4$, $b_1\le b_2$ and $a_i$'s $\in \{1,2,3\}$, $b_i$'s $\in \{1,2,4\}$. We also determine formulas for the number of representations of a positive integer by the quadratic forms $(x_1^2+x_1x_2+x_2^2) + c_1(x_3^2+x_3x_4+x_4^2) + c_2(x_5^2+x_5x_6+x_6^2) + c_3(x_7^2+x_7x_8+x_8^2)$, where $c_1,c_2,c_3\in \{1,2,4,8\}$, $c_1\le c_2\le c_3$.