Gauss's circle problem

In mathematics, the Gauss circle problem is the problem of determining how many integer lattice points there are in a circle centred at the origin and with radius r. 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.

Consider a circle in R² with centre at the origin and radius r ≥ 0. Gauss' circle problem asks how many points there are inside this circle of the form (m,n) where m and n are both integers. Since the equation of this circle is given in Cartesian coordinates by x² + y² = r ², the question is equivalently asking how many pairs of integers m and n there are such that

If the answer for a given r is denoted by N(r) then the following list shows the first few values of N(r) for r an integer between 0 and 12 followed by the list of values ${\displaystyle \pi r^{2}}$ rounded to the nearest integer:

N(r) is roughly πr², the area inside a circle of radius r. 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, πr². So it should be expected that

for some error term E(r) of relatively small absolute value. Finding a correct upper bound for |E(r)| is thus the form the problem has taken. Note that r need not be an integer. After ${\displaystyle N(4)=49}$ one has ${\displaystyle N({\sqrt {17}})=57,N({\sqrt {18}})=61,N({\sqrt {20}})=69,N(5)=81.}$ At these places ${\displaystyle E(r)}$ increases by ${\displaystyle 8,4,8,12}$ after which it decreases (at a rate of ${\displaystyle 2\pi r}$) until the next time it increases.

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