Geodesic
Some analogies of the Hardy--Littlewood equation
Zapiski Nauchnykh Seminarov POMI, Integral lattices and finite linear groups, Tome 116 (1982), pp. 86-95

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We derive an asymptotic expansion for the number of representations of an integer $\mathscr N$ in the form $$ \mathscr N=\ell_1(p,q)+\ell_2(x,y), $$ where $p,q$ are odd primes, $x,y$ are integers, $\ell _1$ and $\ell_2$ are arbitrary primitive quadratic forms with negative discriminant. The equation $\mathscr N=p^2+q^2+x^2+y^2$ was studied earlier by V. A. Plaksin (RZhMat, 1981, 8A135) who used the methods of C. Hooley (RZhMat, 1958, 5451) and Linnik's dispersion method. The author follows Hooley without the use of the dispersion method. The proof is relatively simple. 
@article{ZNSL_1982_116_a8,
     author = {F. B. Koval'chik},
     title = {Some analogies of the {Hardy--Littlewood} equation},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {86--95},
     publisher = {mathdoc},
     volume = {116},
     year = {1982},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1982_116_a8/}
}
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F. B. Koval'chik. Some analogies of the Hardy--Littlewood equation. Zapiski Nauchnykh Seminarov POMI, Integral lattices and finite linear groups, Tome 116 (1982), pp. 86-95. http://geodesic.mathdoc.fr/item/ZNSL_1982_116_a8/