The Gauss circle problem concerns the value of the function $N(r)$, which counts the number of lattice points $(m,n)$ satisfying the inequality $m^2 + n^2 \leq r^2$, that is, lies in the circle centered at the origin with radius $r$. The main term is rather easy to obtain, and the rub lies in estimating the error term. If we write $$\displaystyle N(r) = \pi r^2 + E(r),$$ then Gauss proved that $E(r) = O(r)$, and it is known that $E(r) \ne o(r^\delta)$ for any $\delta \leq 1/2$. Currently, the best established bound, due to Huxley, is of the form $$\displaystyle E(r) = O(r^{131/208}).$$
In 2007, Sylvain Cappell alleged a proof (via an upload to arXiv) that claimed to establish the bound $$\displaystyle E(r) = O(r^{1/2 + \epsilon}),$$ which is best possible. To date, it seems that this proof is still being vetted.
Does anyone have an update on the situation of this proof?
