Certain analogues of the Hardy–Litlewood problem and density methods

Applying density methods of the theory of the Dirichlet $L$-functions, one finds an asymptotic formula for the number of solutions of the equations of the type $N=\varphi(x,y)+m$ and $N=m-\varphi(x,y)$, where $\varphi(x,y)$ is a positive primitive quadratic form, while $m$ is representable by a sum of two squares and runs through its values without repetition.