Estimates for representation numbers of quadratic forms

Let f be a primitive positive integral binary quadratic form of discriminant - D, and let r(f) (n) be the number of representations of n by f up to automorphisms of f. In this article, we give estimates and asymptoticsfor the quantity Sigma(n <= x) r(f) (n)beta for all beta >= 0 and uniformly in D = o(x). As a consequence, we get more-precise estimates for the number of integers which can be written as the sum of two powerful numbers.