Number of Lattice Points in Circle

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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:

$\map d n = \map {\mathcal O} {n^{1/4 + \epsilon} } = \map {\mathcal O} {n^{\theta} }$


Progress

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

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

The lower bound $\dfrac 1 4$ was established by Godfrey Harold Hardy and Edmund Georg Hermann Landau.


Sources