Number of integral points that are interior to a circle

Question

Find the number of integral points that are interior to the circle $x^{2}+y^2=n^2$.

I could count the number of points using a graph for $n=4$, but what should be the approach for large values of $n?$

Answer 1

The number of integral points is close to the area of the circle, i.e. $\pi n^2$. How close is an interesting question: the bound $O(n)$ can be proved in an elementary fashion, the bound $O(n^{2/3})$ is classically derived from Bessel functions and Fourier inversion (Voronoi). Additionally, since $\mathbb{Z}[i]$ is an Euclidean domain, hence a UFD, the following function

$$ r_2(m) = \left|\{(x,y)\in\mathbb{Z}^2: x^2+y^2=m\}\right| $$ is four times a multiplicative function, namely $$ r_2(m) = 4\sum_{d\mid m}\chi_4(d),\qquad\chi_4(d)=\left\{\begin{array}{rcl}1 &\text{if}& d\equiv 1\pmod{4}\\ -1&\text{if}& d\equiv 3\pmod{4}\\0&\text{if}& d\equiv 0\pmod{2}\end{array}\right. $$ In particular, the number of integral points in $x^2+y^2\leq n^2$ is both $$ \sum_{h=-n}^{n}1+2\left\lfloor\sqrt{n^2-h^2}\right\rfloor = 1+\sum_{m=1}^{n^2}4\sum_{d\mid m}\chi_4(d)$$ and this is approximately $\pi n^2$. See Gauss circle problem.