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  "body": "\n## Summary\n\nA Kloosterman sum is an [[exponential sum|97PBMQCPPM]] over the invertible residues $x$ modulo $m$ with phase $(ax + b\\bar{x})/m$.\nHere $\\bar{x}$ is the inverse of $x$ modulo $m$.\nIt is a real number, and for a prime modulus $p$ not dividing $ab$ Weil's bound gives $\\abs{K(a, b; p)} \\le 2\\sqrt{p}$.\nKloosterman sums appear in the Fourier coefficients of Poincaré series, in the circle method for quadratic forms, and in the spectral theory of automorphic forms.\n\n## Explanations\n\n### L1 human\n\nPair each invertible residue $x$ modulo $m$ with its inverse $\\bar{x}$, and turn $ax + b\\bar{x}$ into a point on the unit circle.\nThe Kloosterman sum adds these points.\nFor a prime modulus $p$ there are $p - 1$ terms, but the inverse map scrambles them so much that the sum has size at most $2\\sqrt{p}$.\nFor $m = 5$ and $a = b = 1$, the sum equals $2 + 2\\cos(4\\pi/5) \\approx 0.38$.\n",
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