For $r > 0$, let $L(r) = \# \{ (x,y) \in \mathbb{Z}^2 \ | \ x^2 + y^2 \leq r^2\}$ be the number of lattice points lying on or inside the standard circle of radius $r$. It is easy to see that $L(r) \sim \pi r^2$ as $r \rightarrow \infty$. The Gauss circle problem is to give the best possible error bounds: put
$E(r) = |L(r) – \pi r^2|$.
Gauss himself gave the elementary bound $E(r) = O(r)$. In 1916 Hardy and Landau showed that it is not the case that $E(r) = O(r^{\frac{1}{2}})$. It is now believed that this is "almost" true: i.e.:
Gauss Circle Conjecture: For every $\epsilon > 0$, $E(r) = O_{\epsilon}(r^{\frac{1}{2}+\epsilon})$.
So far as I know the best published result is a 1993 theorem of Huxley, who shows one may take $\epsilon > \frac{19}{146}$.
(For a little more information, see here, Wayback Machine.)
In early 2007 I was teaching an elementary number theory class when I noticed that Cappell and Shaneson had uploaded a preprint to the arxiv claiming to prove the Gauss Circle Conjecture:
Two more versions were uploaded, the last in July of 2007.
It is now a little more than three years later, and so far as I know the paper has neither been published nor retracted. This seems like a strange state of affairs for an important classical problem. Can someone say what the status of the Gauss Circle Problem is today? Is the argument of Cappell and Shaneson correct? Or is there a known flaw?
Best Answer
When Cappell visited UWM a few months ago, one of my colleagues asked him about the status of the paper. The answer was that it is "still in works", which in plain English, probably, means "having severe problems with some remote hope to fix them". The point is that it contains no idea that hadn't been well-known to experts before it appeared, just an enormous amount of "brute forcing" (which, by the way, makes it very hard to read). Sometimes you can succeed by being just more persistent than others but it doesn't seem to be the case here. The consensus is that the existing methods have already been brought to their extreme and to proceed some fresh idea is required.