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What is the status of the Gauss Circle Problem?
The Gauss circle problem is the following: Let $N(r)$ denote the number of solutions in integer pairs $(i,j)$ to the inequality $i^2 + j^2 \leq r^2$ (namely, the number of lattice points on or inside the disc centered at 0 with radius $r$). Then it is easy to see we should have $N(r) = \pi r^2 + E(r)$ for some error term $E(r)$, where the key is to estimate $E(r)$. Gauss proved that $E(r) \leq 2\sqrt{2}\pi r$, and Landau showed that $E(r) \ne o(r^{1/2}\log^{1/4}(r))$. The conjecture is that $E(r) = O(r^{1/2 + \epsilon})$ for any $\epsilon > 0$. If the conjecture is true, then the squares will provide an explicit example of a subset of positive integers $A$ such that that the representation function $r_A(n)$ defined to be the number of ways of writing $n$ as a sum of two elements of $A$ satisfies $\displaystyle \sum_{j \leq n} r_A(j) = cn + O(n^{1/4}\log(n))$. Such sets $A$ exist by a result of I. Ruzsa in 1999, but no known examples exist.
So my question is, what is the best known result on the Gauss circle problem? Or in lieu of that, a good explanation on why this problem is so difficult?
