The Additive Problem for the Number of Representations as a Sum of Two Squares

We improve a previous unconditional result about the asymptotic behavior of $$\sum _{n\le x} r(n)r(n+m)$$ ∑ n ≤ x r ( n ) r ( n + m ) with r ( n ) the number of representations of n as a sum of two squares when m may vary with x .