WOLFRAM|DEMONSTRATIONS PROJECT

Gauss's Circle Problem

r

5.

N(r) = π

2

r

+0.492037r

Count the number of lattice points

N(r)

inside the boundary of a circle of radius

r

with center at the origin. Then

N(r)

can be written

N(r)π

2

r

+E(r)

, where Gauss showed that

E(r)≤2

2

πr≈8.8858r

.