Representations by octonary quadratic forms with coefficients $1$, $2$, $3$ or $6$

TL;DR

This paper uses modular forms to derive formulas for counting how many positive integers can be represented by specific octonary quadratic forms with coefficients 1, 2, 3, or 6.

Contribution

It provides explicit formulas for representation counts of integers by certain octonary quadratic forms using modular form techniques.

Findings

Formulas for the number of representations are explicitly derived.

The results apply to forms with coefficients 1, 2, 3, and 6.

The approach advances understanding of quadratic forms via modular forms.

Abstract

Using modular forms we determine formulas for the number of representations of a positive integer by diagonal octonary quadratic forms with coefficients $1$, $2$, $3$ or $6$.