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:
- $\map d n = \map \OO {n^{1/4 + \epsilon} } = \map \OO {n^{\theta} }$
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
- 1963: Chen Jingrun: The Lattice Points in the Circle (Sci. Sinica Vol. 4, no. 2: pp. 322 – 339)
- 1983: François Le Lionnais and Jean Brette: Les Nombres Remarquables ... (previous) ... (next): $1/4 \le \theta \le 12/37$