points lie on or inside the circle. It is not hard to prove that the answer is , where as . Again, the difficult part and a great achievement of analytic number theory is obtaining specific upper bounds on the error term E(r).

It was shown by Gauss that . In general, an O(r) error term would be possible with the unit circle (or, more properly, the closed unit disk) replaced by the dilates of any bounded planar region with piecewise smooth boundary. Furthermore, replacing the unit circle by the unit square, the error term for the