Geodesic
Certain analogues of the Hardy--Litlewood problem and density methods
Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 4, Tome 112 (1981), pp. 121-142

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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. 
@article{ZNSL_1981_112_a9,
     author = {F. B. Koval'chik},
     title = {Certain analogues of the {Hardy--Litlewood} problem and density methods},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {121--142},
     publisher = {mathdoc},
     volume = {112},
     year = {1981},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1981_112_a9/}
}
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F. B. Koval'chik. Certain analogues of the Hardy--Litlewood problem and density methods. Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 4, Tome 112 (1981), pp. 121-142. http://geodesic.mathdoc.fr/item/ZNSL_1981_112_a9/