The Hardy-Littlewood problem (Q1589170)

Hardy and Littlewood solved the problem of the representation of a natural number as the sum of two squares and a prime. They obtained an asymptotic formula for the corresponding number of such representations. At present various generalizations of the result are known. This paper gives an asymptotic formula for the number of representations of the natural number \(N\) as a sum \[ N=x^2+y^2+n, \] where \(x,y \in \mathbb{Z}\), \(n\) is a given number having \(k\) prime factors from arithmetic progressions. The modulo of the progressions and the factors may be restricted to some conditions.