{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "7VWGA444K3"
    },
    {
      "field": "uses",
      "target": "C01Z9QRGZY"
    },
    {
      "field": "defines",
      "target": "RGT9HH7SQQ"
    }
  ],
  "body": "\n## Summary\n\nThe circle method is an analytic technique for counting the solutions of additive equations.\nExamples are writing $N$ as a sum of $s$ $k$-th powers, or as a sum of three primes.\nIt writes the number of solutions as an integral over the unit circle of a power of an [[exponential sum|97PBMQCPPM]].\nThe integral is split into major arcs near rationals with small denominators, which give the main term.\nThe rest, the minor arcs, are shown to contribute less.\nIt goes back to Hardy and Ramanujan's work on the partition function and was developed by Hardy and Littlewood for [[Waring's problem|7VWGA444K3]].\n\n## Explanations\n\n### L1 human\n\nSay you want to count the ways of writing $N$ as a sum of three cubes.\nOne builds a sum of waves, one wave for each cube, and raises it to the third power.\nIntegrating against a wave of frequency $N$ picks out exactly the combinations that add up to $N$.\nNear simple fractions such as $0$, $1/2$ or $1/3$ the waves line up and the integrand is large; this gives the main term.\nEverywhere else the waves mostly cancel, and the hard work is to prove that this cancellation is strong enough.\nFor example, this is how Vinogradov proved that every large odd number is a sum of three primes.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "kind": "concept",
    "msc": [
      "11P55",
      "11P05",
      "11P32"
    ],
    "name": "Hardy–Littlewood circle method",
    "same_as": {
      "wikidata": "Q2543359",
      "wikipedia": "Hardy–Ramanujan–Littlewood circle method"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Hardy–Ramanujan–Littlewood_circle_method"
      }
    ],
    "tag": "ZQFBEX62D0"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/ZQFBEX62D0/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/ZQFBEX62D0.md",
  "msc": [
    "11P55",
    "11P05",
    "11P32"
  ],
  "name": "Hardy–Littlewood circle method",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/ZQF/ZQFBEX62D0.md",
  "status": null,
  "tag": "ZQFBEX62D0",
  "uri": "m2://ZQFBEX62D0",
  "url": "https://math2.tamnd.com/wiki/hardy-littlewood-circle-method/",
  "writing": {
    "tag": "ZQFBEX62D0",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}