How many integer lattice points there are in a circle
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 formula_1. 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.
Consider a circle in formula_2 with center at the origin and radius formula_3. Gauss's circle problem asks how many points there are inside this circle of the form formula_4 where formula_5 and formula_6 are both integers. Since the equation of this circle is given in Cartesian coordinates by formula_7, the question is equivalently asking how many pairs of integers "m" and "n" there are such that
formula_8
If the answer for a given formula_1 is denoted by formula_10 then the following list shows the first few values of formula_10 for "formula_1" an integer between 0 and 12 followed by the list of values formula_13 rounded to the nearest integer:
1, 5, 13, 29, 49, 81, 113, 149, 197, 253, 317, 377, 441 (sequence in the OEIS) 
0, 3, 13, 28, 50, 79, 113, 154, 201, 254, 314, 380, 452 (sequence in the OEIS)
Bounds on a solution and conjecture.
formula_10 is roughly formula_15, the area inside a circle of radius formula_1. This is because on average, each unit square contains one lattice point. Thus, the actual number of lattice points in the circle is approximately equal to its area, formula_15. So it should be expected that
formula_18
for some error term formula_19 of relatively small absolute value. Finding a correct upper bound for formula_20 is thus the form the problem has taken. Note that formula_1 does not have to be an integer. After formula_22 one hasformula_23 At these places formula_24 increases by formula_25 after which it decreases (at a rate of formula_26) until the next time it increases.
Gauss managed to prove that
formula_27
Hardy and, independently, Landau found a lower bound by showing that
formula_28
using the little o-notation. It is conjectured that the correct bound is
formula_29
Writing formula_30, the current bounds on formula_31 are
formula_32
with the lower bound from Hardy and Landau in 1915, and the upper bound proved by Martin Huxley in 2000.
Exact forms.
The value of formula_10 can be given by several series. In terms of a sum involving the floor function it can be expressed as:
formula_34
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 formula_35 is defined as the number of ways of writing the number formula_6 as the sum of two squares. Then
formula_37
Most recent progress rests on the following Identity, which has been first discovered by Hardy:
formula_38
where formula_39 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 integer solutions formula_40 to the inequality
formula_41
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, standing in the origin. If the number of such solutions is denoted formula_42 then the values of formula_42 for "formula_1" taking small integer values are 
0, 4, 8, 16, 32, 48, 72, 88, 120, 152, 192 … (sequence in the OEIS).
Using the same ideas as the usual Gauss circle problem and the fact that the probability that two integers are coprime is formula_45, it is relatively straightforward to show that
formula_46
As with the usual circle problem, the problematic part of the primitive circle problem is reducing the exponent in the error term. At present, the best known exponent is formula_47 if one assumes the Riemann hypothesis. Without assuming the Riemann hypothesis, the best upper bound currently known is
formula_48
for a positive constant formula_49. In particular, no bound on the error term of the form formula_50 for any formula_51 is currently known that does not assume the Riemann Hypothesis.
Notes.
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