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  "body": "\n## Statement\n\nLet $\\zeta$ be the [[Riemann zeta function|1XWBY4N0VT]].\nFor every complex number $s \\ne 1$ with $\\operatorname{Re} s \\ge 1$, one has $\\zeta(s) \\ne 0$.\nIn particular $\\zeta(1 + it) \\ne 0$ for every real $t \\ne 0$.\n\n## Remarks\n\nFor $\\operatorname{Re} s \u003e 1$ this follows from the [[Euler product formula|0C3PN4075R]], since a convergent product of nonzero factors is nonzero.\nFor the line $\\operatorname{Re} s = 1$, the classical argument uses the inequality $3 + 4\\cos\\theta + \\cos 2\\theta = 2(1 + \\cos\\theta)^2 \\ge 0$.\nWith the Euler product it gives $\\abs{\\zeta(\\sigma)^3 \\, \\zeta(\\sigma + it)^4 \\, \\zeta(\\sigma + 2it)} \\ge 1$ for $\\sigma \u003e 1$.\nIf $\\zeta(1 + it) = 0$ with $t \\ne 0$, then as $\\sigma \\to 1^+$ the left side behaves like $(\\sigma - 1)^{-3} (\\sigma - 1)^{4} \\to 0$, because $\\zeta$ has a simple pole at $1$ and $\\zeta(1 + 2it)$ is finite; this is a contradiction.\n\n## Explanations\n\n### L1 human\n\nThe zeros of the zeta function control how regularly the primes are spread.\nThis theorem says that no zero lies on the line $\\operatorname{Re} s = 1$, the edge of the region where the series converges.\nIt is the key analytic input in the proof of the prime number theorem.\n\n### L2 human\n\nThe [[prime number theorem|0EHBY5Y84N]] $\\pi(x) \\sim x / \\log x$ follows from this non-vanishing by a Tauberian theorem, such as the Wiener–Ikehara theorem, applied to $-\\zeta'(s)/\\zeta(s) = \\sum_n \\Lambda(n) n^{-s}$.\nHadamard and de la Vallée Poussin proved the prime number theorem this way in 1896.\nQuantitative zero-free regions, such as $\\sigma \u003e 1 - c/\\log(\\abs{t} + 2)$, give the prime number theorem with an error term.\nThe same method shows $L(1 + it, \\chi) \\ne 0$ for [[Dirichlet L-functions|9D2Z543EJQ]].\n",
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  "name": "Non-vanishing of the zeta function on the line Re s = 1",
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