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  "body": "\n## Problem\n\nFor a real number $r \u003e 0$, let $N(r)$ be the number of points $(x, y) \\in \\Z^2$ with $x^2 + y^2 \\le r^2$, and let $E(r) = N(r) - \\pi r^2$.\nThe problem is to determine the infimum $\\theta$ of the real numbers $\\alpha$ such that $E(r) = O(r^{\\alpha})$ as $r \\to \\infty$, in [[big O notation|0TGV80E6A4]].\n\n## Remarks\n\nGauss observed that $\\abs{E(r)} \\le 2\\sqrt{2}\\,\\pi r$ for $r \\ge 1$: the unit squares centred at the counted points lie between the discs of radius $r - \\sqrt2/2$ and $r + \\sqrt2/2$.\nSo $\\theta \\le 1$.\nSierpiński proved in 1906 that $E(r) = O(r^{2/3})$.\nHardy and Landau showed independently that $E(r)$ is not $O(r^{1/2})$, so $\\theta \\ge 1/2$.\nIt is conjectured that $\\theta = 1/2$, that is, $E(r) = O(r^{1/2 + \\varepsilon})$ for every $\\varepsilon \u003e 0$.\nMartin Huxley proved $E(r) = O(r^{131/208 + \\varepsilon})$ in 2003, and the problem remains open.\nFor example, $N(1) = 5$ and $N(2) = 13$, while $\\pi \\approx 3.14$ and $4\\pi \\approx 12.57$.\n\n## Explanations\n\n### L1 human\n\nDraw a circle of radius $r$ around the origin on squared paper and count the grid points inside it.\nThe count is close to the area $\\pi r^2$, since each grid point owns one unit square.\nThe question is how big the error can be.\nThe easy bound is about the length of the circle, but the true error appears to be much smaller, about the square root of that.\n",
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