Gauss circle problem

Remarks

Gauss observed that $\abs{E(r)} \le 2\sqrt{2}\,\pi r$ for $r \ge 1$: the unit squares centred at the counted points lie between the discs of radius $r - \sqrt2/2$ and $r + \sqrt2/2$. So $\theta \le 1$. Sierpiński proved in 1906 that $E(r) = O(r^{2/3})$. Hardy and Landau showed independently that $E(r)$ is not $O(r^{1/2})$, so $\theta \ge 1/2$. It is conjectured that $\theta = 1/2$, that is, $E(r) = O(r^{1/2 + \varepsilon})$ for every $\varepsilon > 0$. Martin Huxley proved $E(r) = O(r^{131/208 + \varepsilon})$ in 2003, and the problem remains open. For example, $N(1) = 5$ and $N(2) = 13$, while $\pi \approx 3.14$ and $4\pi \approx 12.57$.