On the average orders of the error term in the circle problem

For a natural number n, let r(n) denote the number of ways of writing eta as a sum. of two squares, and P(x) the remainder term in the circle problem of Gauss, that is, P(x) = Sigma(x <= n), r(n) - pi x. The purpose of this paper is to study some properties of the summatory function Sigma(n <= x) P(n)(k) with an arbitrarily fixed natural number k. In particular, we consider the cases k = 2 and 3 in detail.