The multiplication table constant and sums of two squares

Let r_0(n) be the number of representations of n as a sum of two squares, and r_1(n) count the number of representations of n as a sum of an integer square and a prime square. The asymptotic formulas for the moments of r_0(n), with k greater than 1 summed over n up to x are well-known via classical complex analytic approach. As for the moments of r_1(n), the only cases with known asymptotics are k=0,1,2. For k=2 the main term comes from both trivial solutions (i.e. those with a^2+p^2=b^2+q^2 implying p=q) and the non-trivial ones.  The last ones lie on a very thin set of extremely large values of r_1 and we are interested in getting better understanding of this set. We will show that the number of integers n up to x such that r_1(n) is greater than 1 equals pi/2 * x/ log x minus a secondary term of smaller size with a saving of (log x)^delta, where delta = 1 - (1+loglog 2)/log 2 = 0.086... is the Erdos-Tenenbaum-Ford constant. This, in particular, implies that the main contribution to the sum of r_1^2(n) comes from those integers n with  \omega(n) ~ 2log log x and  r_1(n)= (log x)^{log 4-1+o(1). This is a joint work with Andrew Granville and Cihan Sabuncu