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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''. The first progress on a solution was made by [[Carl Friedrich Gauss]], hence its name.
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==The problem==
 
Consider a circle in '''R'''<sup>2</sup> with centre at the origin and radius ''r''&nbsp;≥&nbsp;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 coordinate system|Cartesian coordinates]] by ''x''<sup>2</sup>&nbsp;+&nbsp;''y''<sup>2</sup>&nbsp;=&nbsp;''r''<sup>2</sup>, the question is equivalently asking how many pairs of integers ''m'' and ''n'' there are such that
:<math>m^2+n^2\leq r^2.</math>
 
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 10:
:1, 5, 13, 29, 49, 81, 113, 149, 197, 253, 317 {{OEIS|id=A000328}}.
 
==Bounds on a solution and conjecture==
The [[Area of a disk|area inside a circle]] of radius ''r'' is given by π''r''<sup>2</sup>, and since a square of area 1 in '''R'''<sup>2</sup> contains one integer point, the expected answer to the problem could be about π''r''<sup>2</sup>. In fact it should be slightly higher than this, since circles are [[Isoperimetric inequality|more efficient at enclosing space]] than squares.  So in fact it should be expected that
:<math>N(r)=\pi r^2 +E(r)\,</math>
for some error term ''E''(''r''). Finding a correct upper bound for ''E''(''r'') is thus the form the problem has taken.
 
Gauss managed to prove<ref name="Hardy">G.H. Hardy, ''Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed.'' New York: Chelsea, (1999), p.67.</ref> that
:<math>E(r)\leq 2\sqrt{2}\pi r.</math>
[[G. H. Hardy|Hardy]]<ref>G.H. Hardy, ''On the Expression of a Number as the Sum of Two Squares'', Quart. J. Math. '''46''', (1915), pp.263&ndash;283.</ref> and, independently, [[Edmund Landau|Landau]] found a lower bound by showing that
:<math>E(r)\neq o\left(r^{1/2}(\log r)^{1/4}\right),</math>
using the [[Big_O_notation#Little-o_notation|little o-notation]]. It is conjectured<ref name="Guy">R.K. Guy, ''Unsolved problems in number theory, Third edition'', Springer, (2004), pp.365&ndash;366.</ref> that the correct bound is
:<math>E(r)=O\left(r^{1/2+\varepsilon}\right).</math>
 
Writing |''E''(''r'')|&nbsp;≤&nbsp;''Cr''<sup>''t''</sup>, the current bounds on ''t'' are
:<math>\frac{1}{2}< t\leq\frac{131}{208}=0.6298\ldots,</math>
with the lower bound from Hardy and Landau in 1915, and the upper bound proved by [[Martin Huxley|Huxley]] in 2000.<ref>M.N. Huxley, ''Integer points, exponential sums and the Riemann zeta function'', Number theory for the millennium, II (Urbana, IL, 2000) pp.275&ndash;290, A K Peters, Natick, MA, 2002, {{MR|1956254}}.</ref>
 
In 2007, [[Sylvain Cappell]] and Julius Shaneson deposited a paper in the [[arXiv]] claiming to prove the bound of ''O''(''r''<sup>1/2+ε</sup>).<ref name="Cap-Sha">S. Cappell and J. Shaneson, ''Some Problems in Number Theory I: The Circle Problem'', {{arxiv|math/0702613}}, (2007).</ref>
 
==Exact forms==
 
The value of ''N''(''r'') can be given by several series. In terms of a sum involving the [[floor function]] it can be expressed as:<ref>D. Hilbert and S. Cohn-Vossen, ''Geometry and the Imagination'', New York: Chelsea, (1999), pp.37&ndash;38.</ref>
:<math>N(r)=1+4\sum_{i=0}^\infty \left(\left\lfloor\frac{r^2}{4i+1}\right\rfloor-\left\lfloor\frac{r^2}{4i+3}\right\rfloor\right).</math>
 
A much simpler sum appears if the sum of squares function ''r''<sub>2</sub>(''n'') is defined as the number of ways of writing the number ''n'' as the sum of two squares. Then<ref name="Hardy"/>
:<math>N(r)=\sum_{n=0}^{r^2} r_2(n).</math>
 
==Generalisations==
 
Although the original problem asks for integer lattice points in a circle, there is no reason not to consider other shapes or [[conics]], indeed [[Dirichlet's_divisor_problem#Dirichlet's_divisor_problem|Dirichlet's divisor problem]] is the equivalent problem where the circle is replaced by the rectangular [[hyperbola]].<ref name="Guy"/>  Similarly one could extend the question from two dimensions to higher dimensions, and ask for integer points within a [[N-sphere|sphere]] or other objects.  If one ignores the geometry and merely considers the problem an algebraic one of [[Diophantine inequality|Diophantine inequalities]] then there one could increase the exponents appearing in the problem from squares to cubes, or higher.
 
===The primitive circle problem===
 
Another generalisation is to calculate the number of [[coprime]] integer solutions ''m'', ''n'' to the equation
:<math>m^2+n^2\leq r^2.\,</math>
This problem is known as the '''primitive circle problem''', as it involves searching for primitive solutions to the original circle problem.<ref name="jw02">J. Wu, ''On the primitive circle problem'', Monatsh. Math. '''135''' (2002), pp.69&ndash;81.</ref> If the number of such solutions is denoted ''V''(''r'') then the values of ''V''(''r'') for ''r'' taking small integer values are
:0, 4, 8, 16, 32, 48, 72, 88, 120, 152, 192 … {{OEIS|A175341}}.
Using the same ideas as the usual Gauss circle problem and the fact that the [[Coprime#Probabilities|probability that two integers are coprime]] is 6/''π''<sup>2</sup>, it is relatively straightforward to show that
:<math>V(r)=\frac{6}{\pi}r^2+O(r^{1+\varepsilon}).</math>
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 221/304&nbsp;+&nbsp;''ε'' if one assumes the [[Riemann hypothesis]].<ref name="jw02" /> Without assuming the Riemann hypothesis, the best known upper bound is
:<math>V(r)=\frac{6}{\pi}r^2+O(r\exp(-c(\log r)^{3/5}(\log\log r^2)^{-1/5}))</math>
for a positive constant ''c''.<ref name="jw02" /> In particular, no bound on the error term of the form 1&nbsp;&minus;&nbsp;''ε'' for any ''ε''&nbsp;>&nbsp;0 is currently known that does not assume the Riemann Hypothesis.
 
==Notes==
{{reflist}}
 
==External links==
*{{MathWorld|urlname=GausssCircleProblem|title=Gauss's circle problem}}
 
[[Category:Arithmetic functions]]
[[Category:Lattice points]]
[[Category:Unsolved problems in mathematics]]

Latest revision as of 10:58, 21 November 2014

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