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  "body": "\n## Definition\n\nFor a complex number $s$ and an integer $n \\ge 1$, let $n^{-s} = e^{-s \\log n}$.\nFor $\\operatorname{Re} s \u003e 1$, the [[Dirichlet series|8MWTEF6462]]\n$$\\zeta(s) = \\sum_{n=1}^{\\infty} \\frac{1}{n^s}$$\nconverges absolutely, and it defines a [[holomorphic function|0NAHXR1FRH]] on the half-plane $\\operatorname{Re} s \u003e 1$.\nThis function has a unique [[analytic continuation|0M3YQJ81NB]] to a [[meromorphic function|0ZZ7ZYDR8W]] on $\\C$.\nThe *Riemann zeta function* is this meromorphic function $\\zeta$.\nIts only pole is a simple pole at $s = 1$, with residue $1$.\n\n## Remarks\n\nExample: $\\zeta(2) = \\pi^2/6$, the value of the series in the [[Basel problem|0DMY75D6FV]].\nNon-example of the series form: at $s = 1$ the series is the harmonic series, which diverges, and $\\zeta$ has its pole there.\nAt $s = 0$ or $s = -1$ the series diverges too, but $\\zeta$ is defined there by continuation: $\\zeta(0) = -1/2$ and $\\zeta(-1) = -1/12$.\nFor $\\operatorname{Re} s \u003e 0$ the continuation is given by\n$$\\zeta(s) = \\frac{s}{s - 1} - s \\int_1^{\\infty} \\frac{x - \\floor{x}}{x^{s+1}} \\, dx,$$\nsince the integral converges there.\nThe [[functional equation|NFXKNDJX9C]] gives the continuation to all of $\\C$.\nThe uniqueness of the continuation follows from the [[identity theorem|0YBVKMYX0W]].\n",
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