Editing Analytic number theory (section)

=== Diophantine problems ===
{{main|Diophantine problem}}

[[Diophantine problem]]s are concerned with integer solutions to polynomial equations: one may study the distribution of solutions, that is, counting solutions according to some measure of "size" or ''[[height function|height]]''.

An important example is the [[Gauss circle problem]], which asks for integers points (''x''&nbsp;''y'') which satisfy
:<math>x^2+y^2\leq r^2.</math>
In geometrical terms, given a circle centered about the origin in the plane with radius ''r'', the problem asks how many integer lattice points lie on or inside the circle. It is not hard to prove that the answer is <math>\pi r^2 + E(r)</math>, where <math>E(r)/r^2 \to 0</math> as <math>r \to \infty</math>.  Again, the difficult part and a great achievement of analytic number theory is obtaining specific upper bounds on the error term&nbsp;''E''(''r'').

It was shown by Gauss that <math> E(r) = O(r)</math>. In general, an ''O''(''r'') error term would be possible with the unit circle (or, more properly, the closed unit disk) replaced by the dilates of any bounded planar region with piecewise smooth boundary.  Furthermore, replacing the unit circle by the unit square, the error term for the general problem can be as large as a linear function of&nbsp;''r''.  Therefore, getting an [[error bound]] of the form <math>O(r^{\delta})</math>
for some <math>\delta < 1</math> in the case of the circle is a significant improvement.  The first to attain this was
[[Wacław Sierpiński|Sierpiński]] in 1906, who showed <math> E(r) = O(r^{2/3})</math>.  In 1915, Hardy and [[Edmund Landau|Landau]] each showed that one does ''not'' have <math>E(r) = O(r^{1/2})</math>.  Since then the goal has been to show that for each fixed <math>\epsilon > 0</math> there exists a real number <math>C(\epsilon)</math> such that <math>E(r) \leq C(\epsilon) r^{1/2 + \epsilon}</math>.

In 2000 [[Martin Huxley|Huxley]] showed<ref>M.N. Huxley, ''Integer points, exponential sums and the Riemann zeta function'', Number theory for the millennium, II (Urbana, IL, 2000) pp.275–290, A K Peters, Natick, MA, 2002, {{MR|1956254}}.</ref> that <math>E(r) = O(r^{131/208})</math>, which is the best published result.