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  "body": "\n## Statement\n\nLet $\\Gamma$ be a [[congruence subgroup|1H6R856P3C]] of the [[modular group|VEKV1EM77B]] $\\mathrm{SL}_2(\\Z)$, acting on the [[upper half-plane|X6XGGH5FPW]] $\\mathbb H$.\nLet $\\Delta = -y^2 (\\partial_x^2 + \\partial_y^2)$ be the hyperbolic [[Laplace–Beltrami operator|F4NWZNTRMD]] on $L^2(\\Gamma \\backslash \\mathbb H)$.\nThe conjecture states that every nonzero [[eigenvalue|0BQAA2BK3Q]] $\\lambda$ of $\\Delta$ on $L^2(\\Gamma \\backslash \\mathbb H)$ satisfies $\\lambda \\ge \\tfrac14$.\n\n## Remarks\n\nThe eigenfunctions in question are the [[Maass cusp forms|3R959YSNYP]] for $\\Gamma$, since the continuous spectrum is $[\\tfrac14, \\infty)$.\nSelberg proved the bound $\\lambda \\ge \\tfrac{3}{16}$.\nThe best bound known is $\\lambda \\ge \\tfrac{975}{4096}$, due to Kim and Sarnak.\nThe conjecture is known for $\\mathrm{SL}_2(\\Z)$ itself and for congruence subgroups of small level.\nIt is the archimedean case of the generalised Ramanujan conjecture for $\\mathrm{GL}_2$: an eigenvalue $\\lambda = \\tfrac14 - s^2$ with $0 \u003c s \u003c \\tfrac12$ would give a non-tempered local component at the real place.\nThe arithmetic hypothesis matters: for general cofinite Fuchsian groups, eigenvalues below $\\tfrac14$ can occur.\n\n## Explanations\n\n### L1 human\n\nThe quotient of the upper half-plane by a congruence group is a surface with a few cusps, and it can vibrate at certain frequencies.\nSelberg predicted that, apart from the trivial vibration, all frequencies are at least $\\tfrac14$, the bottom of the continuous spectrum.\nThis means that these arithmetic surfaces are very well connected, with no slow vibration, and it has consequences for counting problems and for expander graphs.\n",
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    "about": [
      "F4NWZNTRMD",
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    "history": {
      "first_stated": "1965",
      "posed_by": "Atle Selberg"
    },
    "kind": "conjecture",
    "msc": [
      "11F72",
      "11F12",
      "58J50"
    ],
    "name": "Selberg's 1/4 conjecture",
    "same_as": {
      "wikidata": "Q7447522",
      "wikipedia": "Selberg's 1/4 conjecture"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Selberg%27s_1/4_conjecture"
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      "F4NWZNTRMD",
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  "link": "https://tamnd.github.io/math2/node/B7K7NY8NTJ/",
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      "field": "about",
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  "markdown": "https://tamnd.github.io/math2/node/B7K7NY8NTJ.md",
  "msc": [
    "11F72",
    "11F12",
    "58J50"
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  "name": "Selberg's 1/4 conjecture",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/B7K/B7K7NY8NTJ.md",
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  "tag": "B7K7NY8NTJ",
  "uri": "m2://B7K7NY8NTJ",
  "url": "https://tamnd.github.io/math2/wiki/selbergs-1-4-conjecture/",
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    "tag": "B7K7NY8NTJ",
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    "level": "W0",
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