Samuel Holmin: The Number of Points in a Ball from a Random Lattice

How many lattice points are there inside a circle of large
radius? It is easy to show that this number is approximated by the area of
the circle. There is a simple heuristic which gives a suggestion for the
size of the error of this approximation, but we do not yet know if it is
true. I will give a bried overview of the history of this problem, and I
will show that the desired bound is attained in the mean square for random
lattices.