Question

A lattice point in the plane is a point (a, b) with both coordinates equal to integers. For example, (-1, 2) is a lattice point but (1/2, 3) is not. If D(R) is the disk of radius R and center the origin, count the lattice points inside D(R) and call this number L(R). What is the limit, limR→∞L(R)/R^2?

Answer 1

Is there anyone who me me?

Answer 2

google turned up a nice article in Wikipedia on this. Gauss showed that
L(R) = r^2 + E(r)
where |E(r)| <= 2√2 πr
So, lim(r→∞) L(r)/r^2 = π

You can read the article, which starts with the idea that

N(r) is roughly πr^2, the area inside a circle of radius r. This is because on average, each unit square contains one lattice point. Thus, the actual number of lattice points in the circle is approximately equal to its area, πr^2