Gauss's circle problem

Gauss's circle problem asks: if you draw a circle with a large radius on a grid of points with integer coordinates (like dots on graph paper), how many points fall inside or on the circle? As the circle gets bigger, the number of these lattice points roughly equals the circle’s area (π times radius squared). The challenge is understanding how the actual count differs from this area estimate—specifically, how the error term behaves as the radius grows. The problem is about finding the most accurate way to measure the difference between the count and the circle’s true area as the radius increases.