# What is the status of the Gauss Circle Problem?

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 $\epsilon > \frac{19}{146}$.

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:

http://arxiv.org/abs/math/0702613

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?

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Cappell's Wikipedia page says that the paper "is still being vetted by experts." This was originally mentioned on 29 April 2008, but it has not been changed since. –  Steve Huntsman Mar 23 '10 at 2:31
@SH: Right, that was almost two years ago. I'm asking for an update from an expert. "Still vetting" is a possible answer, although in that case I'd be interested to know which part is taking so long to check. –  Pete L. Clark Mar 23 '10 at 2:40
I've sent an email to Cappell. I took a few courses from him in the nineties and I think he'll remember me. –  Steve Huntsman Mar 23 '10 at 2:58
I'll talk to Shaneson about it and forward the link. –  Justin Curry Mar 23 '10 at 15:52
Bruce Berndt gave a talk last week at Gainesville that gave history and current status, perhaps not from exactly the same viewpoint as yours of course. See math.ufl.edu/~fgarvan/antc-program/2009-10/mar-focused-week/… –  Will Jagy Apr 1 '10 at 4:50