http://en.wikipedia.org/wiki/Gauss%27s_circle_problem

This formula tells how many integer points are inside or on a boundary of a circle of radius r.

for instance at r = 10 it outputs 317 which is the right answer.

Basically it adds infinitely tall infinitely narrow functions at all the integer points from 0 to the radius in the y direction. Then integrates from 0 to the height of the circle at x for each x from 1 to the radius. This gives all the integer points within the first quadrant of the circle not counting those on x=0 or y=0 lines. Then that is multiplied by four for the four quadrants and add those on the x=0 or y=0 lines and then 1 for the center point.

Now, the Dirac function is a logistic function but it can be considered as the limit of:

as s goes to 0. But smaller and smaller values of s can be used to make the answer my formula gives more exact. This is just if it is required of the solution to be purely algebraic.

The function just above is just a normal curve with mean of j, so it's integral is 1, and you make the deviation smaller and smaller which doesn't change the integral but makes it taller and taller. In the limit it is just infinitely tall at only the point j.

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