On the representation of an integer as the sum of a large and a small square

We continue our study of the different representations of an integer n as a sum of two squares initiated in our final paper written with Paul Erdős [1]. We let r(n) denote the number of representations n = a 2 + b 2 counted in the usual way, that is, with regard to both the order and sign of a and b . We have where x is the non-principal Dirichlet character (mod 4); moreover, r ( n )≥0 if and only if n has no prime factor p ≡ 3 (mod 4) with odd exponent. We define the function b ( n ) on the sequence of representable numbers as the least possible value of | b |—for example, we have b (13) = 2, b (25) = 0, b (65) = 1—and we write here and throughout the paper the star denotes that the sum is restricted to the representable integers. The problem considered here is to find an asymptotic formula for this sum, or, less ambitiously, to determine the order of magnitude of the function B(x) .