Some analogies of the Hardy-Littlewood equation

Abstract

We derive an asymptotic expansion for the number of representations of an integer$\mathcal{N}$ in the form

$$\mathcal{N} = \ell _1 \left( {p, q} \right) + \ell _2 \left( {x, y} \right),$$

where p, q are odd primes, x, y are integers, ℓ₁ and ℓ₂ are arbitrary primitive quadratic forms with negative discriminant. The equation$\mathcal{N}$=p²+q²+x²+y² 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.