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  "body": "\n## Statement\n\nLet $\\zeta$ be the [[Riemann zeta function|1XWBY4N0VT]].\nEvery zero $s$ of $\\zeta$ with $0 \u003c \\operatorname{Re} s \u003c 1$ satisfies $\\operatorname{Re} s = \\frac{1}{2}$.\n\n## Remarks\n\nThe zeros of $\\zeta$ are the trivial zeros $-2, -4, -6, \\dots$ and the nontrivial zeros, which lie in the strip $0 \u003c \\operatorname{Re} s \u003c 1$.\nThis follows from the [[functional equation|NFXKNDJX9C]] and the [[non-vanishing on the line Re s = 1|CXES7915R7]].\nSo the conjecture says that every nontrivial zero lies on the critical line $\\operatorname{Re} s = 1/2$.\nThe first nontrivial zeros are at about $\\frac{1}{2} \\pm 14.1347 i$.\nThe conjecture is open.\nRiemann stated it in his 1859 paper on the number of primes below a given size.\nIt is one of Hilbert's problems of 1900 and one of the Clay Millennium Prize Problems.\nKnown partial results: Hardy proved that infinitely many zeros lie on the critical line, and Selberg proved that a positive proportion of them do.\nAn equivalent form: $\\pi(x) = \\operatorname{li}(x) + O(\\sqrt{x} \\log x)$, where $\\pi$ is the [[prime-counting function|0RQSNVTFVZ]] and $\\operatorname{li}(x) = \\int_2^x dt / \\log t$.\nThe [[generalized Riemann hypothesis|DNKA4WBKDS]] states the same for [[Dirichlet L-functions|9D2Z543EJQ]] and other L-functions.\n\n## Explanations\n\n### L1 human\n\nThe zeta function has infinitely many zeros in a vertical strip of the complex plane.\nRiemann guessed that all of them lie exactly on the middle line of this strip.\nIf true, the primes are distributed as regularly as possible: the number of primes up to $x$ differs from its smooth approximation by roughly $\\sqrt{x}$ only.\n",
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      "first_stated": "1859",
      "posed_by": "Bernhard Riemann"
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    "name": "Riemann hypothesis",
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  "markdown": "https://math2.tamnd.com/node/VCGF9YR36X.md",
  "msc": [
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  "name": "Riemann hypothesis",
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  "source": "https://github.com/tamnd/math2/blob/main/nodes/VCG/VCGF9YR36X.md",
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