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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?
Best Answer
The best reference on the subject is M. Huxley's monograph
MR1420620 (97g:11088) Huxley, M. N.(4-WALC) Area, lattice points, and exponential sums. London Mathematical Society Monographs. New Series, 13. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1996. xii+494 pp. ISBN: 0-19-853466-3 11L07 (11J54 11P21)
The proof of the circle conjecture has been claimed by Cappell/Shaneson, but I haven't read their paper, don't know anyone who has, so cannot comment on its correctness (maybe someone else here can).