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  "body": "\n## Definition\n\nLet $m \\ge 1$ be an integer, and let $a, b \\in \\Z$.\nFor a real $t$, write $e(t) = e^{2\\pi i t}$.\nFor an integer $x$ coprime to $m$, let $\\bar{x}$ be an integer with $x \\bar{x} \\equiv 1 \\pmod{m}$.\nThe *Kloosterman sum* is\n$$K(a, b; m) = \\sum_{\\substack{x \\bmod m \\\\ \\gcd(x, m) = 1}} e\\!\\left(\\frac{a x + b \\bar{x}}{m}\\right),$$\nwhere $x$ runs over a set of representatives of the invertible classes of the [[integers modulo m|03VFZ2WETT]].\n\n## Remarks\n\nThe value does not depend on the choice of representatives or of $\\bar{x}$, since the phase is defined modulo $1$.\nExample: $m = 5$, $a = b = 1$.\nThe pairs $(x, \\bar{x})$ are $(1, 1), (2, 3), (3, 2), (4, 4)$, so the phases are $2/5, 1, 1, 8/5$ and\n$$K(1, 1; 5) = e(2/5) + 2 + e(3/5) = 2 + 2\\cos(4\\pi/5) \\approx 0.382.$$\nNon-example: the sum $\\sum_{x = 0}^{m-1} e((a x + b \\bar{x})/m)$ over all residues is not defined, since $\\bar{x}$ does not exist when $\\gcd(x, m) \u003e 1$.\nThe sum is real, because replacing $x$ by $-x$ conjugates each term.\nIf $\\gcd(a, m) = 1$, then $K(a, b; m) = K(1, ab; m)$.\nFor $b = 0$, the sum $K(a, 0; m)$ is the Ramanujan sum $c_m(a)$.\nIt is an [[exponential sum|97PBMQCPPM]] in which the phase function is $x \\mapsto (ax + b\\bar{x})/m$.\n",
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