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

TL;DR

This paper employs modular forms to derive formulas for counting representations of positive integers by specific classes of eight-variable quadratic forms, expanding understanding of their structure and enumeration.

Contribution

It introduces explicit formulas for representations by certain quadratic forms in eight variables, using modular form theory, which is a novel approach for these classes.

Findings

Derived formulas for forms with coefficients in {1,2,3} and {1,2,4}.

Established formulas for forms with coefficients in {1,2,4,8}.

Enhanced methods for counting representations of integers by complex quadratic forms.

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 $c_1,c_2,c_3\in \{1,2,4,8\}$, $c_1\le c_2\le c_3$.