Number of Lattice Points in Circle

Conjecture
Consider the circle $C$ of radius $\sqrt n$ whose center is at the origin of a cartesian plane.

Let $\map R n$ denote the number of lattice points in $C$

Let $\map d n$ denote the difference between the area of $C$ and $\map R n$:
 * $\map d n = \pi n - \map R n$

It is conjectured that:
 * $\map d n = \map \OO {n^{1/4 + \epsilon} } = \map \OO {n^{\theta} }$

Progress
In $1963$, determined that $\theta \le \dfrac {12} {37}$.

Previous to that, the best estimate was $\theta \le \dfrac {17} {53}$, due to.

The lower bound $\dfrac 1 4$ was established by and.