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  "body": "\n## Statement\n\nLet $\\zeta$ be the [[Riemann zeta function|1XWBY4N0VT]].\nFor every $\\varepsilon \u003e 0$ there is a constant $C_\\varepsilon$ such that\n$$\\abs{\\zeta(\\tfrac12 + it)} \\le C_\\varepsilon \\abs{t}^{\\varepsilon}$$\nfor all real $t$ with $\\abs{t} \\ge 1$.\n\n## Remarks\n\nThe conjecture is open.\nThe [[Riemann hypothesis|VCGF9YR36X]] implies it.\nThe functional equation and the Phragmén–Lindelöf principle give the convexity bound $\\zeta(\\tfrac12 + it) = O(\\abs{t}^{1/4 + \\varepsilon})$.\nEvery improvement of the exponent $1/4$ is called a subconvexity bound; Hardy and Littlewood obtained $1/6 + \\varepsilon$ with Weyl's method, and Bourgain obtained $13/84 + \\varepsilon$.\nAn equivalent form: for every integer $k \\ge 1$ and every $\\varepsilon \u003e 0$,\n$$\\frac{1}{T} \\int_{0}^{T} \\abs{\\zeta(\\tfrac12 + it)}^{2k} \\, dt = O(T^{\\varepsilon}).$$\n\n## Explanations\n\n### L1 human\n\nOn the critical line $\\operatorname{Re} s = 1/2$, the zeta function oscillates and its size grows slowly with the height $t$.\nThe Lindelöf hypothesis says that it grows more slowly than any power of $t$, however small.\nIt is weaker than the Riemann hypothesis but still out of reach.\nIt would imply strong bounds on the moments of zeta and on the number of zeros off the critical line.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
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    "history": {
      "first_stated": "1908",
      "posed_by": "Ernst Lindelöf"
    },
    "kind": "conjecture",
    "msc": [
      "11M06",
      "11M26"
    ],
    "name": "Lindelöf hypothesis",
    "same_as": {
      "wikidata": "Q911528",
      "wikipedia": "Lindelöf hypothesis"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Lindelöf_hypothesis"
      }
    ],
    "tag": "PV82BVXE1A",
    "uses": [
      "1XWBY4N0VT",
      "0TGV80E6A4",
      "VCGF9YR36X"
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  "kind": "conjecture",
  "link": "https://math2.tamnd.com/node/PV82BVXE1A/",
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      "target": "VCGF9YR36X"
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  "markdown": "https://math2.tamnd.com/node/PV82BVXE1A.md",
  "msc": [
    "11M06",
    "11M26"
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  "name": "Lindelöf hypothesis",
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  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/PV8/PV82BVXE1A.md",
  "status": null,
  "tag": "PV82BVXE1A",
  "uri": "m2://PV82BVXE1A",
  "url": "https://math2.tamnd.com/wiki/lindelof-hypothesis/",
  "writing": {
    "tag": "PV82BVXE1A",
    "kind": "conjecture",
    "level": "W0",
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        "what": "a source with a locator",
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        "want": 1
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