{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "KVZ0VZ6W01"
    },
    {
      "field": "uses",
      "target": "PMYDHW00KF"
    }
  ],
  "body": "\n## Summary\n\nThe Dedekind eta function is $\\eta(\\tau) = e^{\\pi i \\tau / 12} \\prod_{n \\ge 1} (1 - e^{2 \\pi i n \\tau})$ on the [[upper half-plane|X6XGGH5FPW]].\nIt is holomorphic and never zero there, and it transforms like a modular form of weight $1/2$, up to $24$-th roots of unity.\nIts $24$-th power is the cusp form $\\Delta$ of weight $12$.\nIts reciprocal is, up to a factor, the generating function of the [[partition function|0CV69WDRXW]].\n\n## Explanations\n\n### L1 human\n\nThe eta function is an infinite product $q^{1/24}(1 - q)(1 - q^2)(1 - q^3) \\cdots$ with $q = e^{2 \\pi i \\tau}$.\nExpanding $1/\\prod (1 - q^n)$ counts integer partitions, so eta is tied to counting partitions.\nUnder $\\tau \\mapsto -1/\\tau$ it changes only by the simple factor $\\sqrt{-i\\tau}$.\nThis hidden symmetry is the source of the Hardy–Ramanujan formula for the number of partitions.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "11F20"
    ],
    "name": "Dedekind eta function",
    "same_as": {
      "wikidata": "Q1182161",
      "wikipedia": "Dedekind eta function"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Dedekind_eta_function"
      }
    ],
    "tag": "MX3CVYMNDV"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/MX3CVYMNDV/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/MX3CVYMNDV.md",
  "msc": [
    "11F20"
  ],
  "name": "Dedekind eta function",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/MX3/MX3CVYMNDV.md",
  "status": null,
  "tag": "MX3CVYMNDV",
  "uri": "m2://MX3CVYMNDV",
  "url": "https://math2.tamnd.com/wiki/dedekind-eta-function/",
  "writing": {
    "tag": "MX3CVYMNDV",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}