On the number of representations of a positive integer as a sum of two binary quadratic forms

Abstract

Let $\mathbb{N}$ denote the set of positive integers and let ${\mathbb{N}}_0=\mathbb{N}\cup \{0\}$. Let $a_1{x}^2+b_{1}xy+c_1{y}^2$ and $a_2{z}^2+b_{2}zt+c_2{t}^2$ be two positive-definite, integral, binary quadratic forms. The number of representations of $n\in {\mathbb{N}}_0$ as a sum of these two binary quadratic forms is $$\begin{array}{rl}& N(a_1,b_1,c_1,a_2,b_2,c_2;n):=\text{ card}\{(x,y,z,t)\in {\mathbb{Z}}^4|\\ & n=a_1{x}^2+b_{1}xy+c_1{y}^2+a_2{z}^2+b_{2}zt+c_2{t}^2\}.\end{array}$$ When $(b_1,b_2)\ne (0,0)$ we prove under certain conditions on $a_1,b_1,c_1,a_2,b_2$ and $c_2$ that $N(a_1,b_1,c_1,a_2,b_2,c_2;n)$ can be expressed as a finite linear combination of quantities of the type $N(a,0,b,c,0,d;n)$ with $a,b,c$ and $d$ positive integers, see Theorem 1.1. Thus, when the quantities $N(a,0,b,c,0,d;n)$ are known, we can determine $N(a_1,b_1,c_1,a_2,b_2,c_2;n)$. This determination is carried out for a number of quaternary quadratic forms $a_1{x}^2+b_{1}xy+c_1{y}^2+a_2{z}^2+b_{2}zt+c_2{t}^2$.