Page "Analytic number theory" Paragraph 26

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 general problem can be as large as a linear function of r. Therefore getting an error bound of the form