In mathematics, the Gauss circle problem is the problem of determining how many integer lattice points there are in a circle centered at the origin and with radius. This number is approximated by the area of the circle, so the real problem is to accurately bound the error term describing how the number of points differs from the area. The first progress on a solution was made by Carl Friedrich Gauss, hence its name.

The problem

are both integers. Since the equation of this circle is given in Cartesian coordinates, the question is equivalently asking how many pairs of integers m and n there are such that

for rounded to the nearest integer:

1. 1, 5, 13, 29, 49, 81, 113, 149, 197, 253, 317, 377, 441

2. 0, 3, 13, 28, 50, 79, 113, 154, 201, 254, 314, 380, 452

Bounds on a solution and conjecture

, the area inside a circle. This is because on average, each unit square. So it should be expected that

of relatively small absolute value) until the next time it increases.

Gauss managed to prove,

Hardy and, independently, Landau found a lower bound by showing that

using the little o-notation. It is conjectured that the correct bound is

are

with the lower bound from Hardy and Landau in 1915, and the upper bound proved by Martin Huxley in 2000.

Exact forms

can be given by several series. In terms of a sum involving the floor function it can be expressed as:

This is a consequence of Jacobi's two-square theorem, which follows almost immediately from the Jacobi triple product.

A much simpler sum appears if the sum of squares function as the sum of two squares. Then

Most recent progress rests on the following Identity, which has been first discovered by Hardy:

denotes the Bessel function of the first kind with order 1.

Generalizations

Although the original problem asks for integer lattice points in a circle, there is no reason not to consider other shapes, for example conics; indeed Dirichlet's divisor problem is the equivalent problem where the circle is replaced by the rectangular hyperbola. Similarly one could extend the question from two dimensions to higher dimensions, and ask for integer points within a sphere or other objects. There is an extensive literature on these problems. If one ignores the geometry and merely considers the problem an algebraic one of Diophantine inequalities, then there one could increase the exponents appearing in the problem from squares to cubes, or higher.

The dot planimeter is physical device for estimating the area of shapes based on the same principle. It consists of a square grid of dots, printed on a transparent sheet; the area of a shape can be estimated as the product of the number of dots in the shape with the area of a grid square.

The primitive circle problem

Another generalization is to calculate the number of coprime to the inequality

This problem is known as the primitive circle problem, as it involves searching for primitive solutions to the original circle problem. It can be intuitively understood as the question of how many trees within a distance of r are visible in the Euclid's orchard for taking small integer values are

1. 0, 4, 8, 16, 32, 48, 72, 88, 120, 152, 192 … .

Using the same ideas as the usual Gauss circle problem and the fact that the probability that two integers are coprime, it is relatively straightforward to show that

if one assumes the Riemann hypothesis. Without assuming the Riemann hypothesis, the best upper bound currently known is

. is currently known that does not assume the Riemann Hypothesis.

Notes

External links