Is there a heuristic argument behind the exponent in the [circle problem](https://en.wikipedia.org/wiki/Gauss_circle_problem)?  The problem that I am referring to is the following:  Consider a circle of radius $R$ centered at the origin in the plane and let $N(R)$ denote the number of integer lattice points contained in the circle.  Then it easy to show that $N(R)/ \pi R^2 \to 1$ as $R \to \infty$.  The circle problem asks what is the optimal exponent for the error term.