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 http://math.uga.edu/~pete/NT2009gausscircle.pdf)

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?