Counting integer solutions to $x^2 + y ^2 < n$

Question

How can I count the number of integer solutions, $\mu(n)$, of $x^2+y^2 < n$, and then hopefully look at behavior as $n \to +\infty$?

Answer 1

The behaviour as $n\to\infty$ is simpler than counting exactly in the finite case: You count exactly the integer points in a circle of radius $\sqrt n$. Since the circle is not too "chaotic", the number of points is essentially the area of the circle: $\mu(n)\sim \pi n$.

Answer 2

This is the Gauss circle problem; you’ll find considerable information there and at MathWorld. The latter has numerous references, and there are more references at OEIS A000328.