{
  "aliases": null,
  "backlinks": [
    {
      "field": "specializes",
      "target": "2767J23XHN"
    },
    {
      "field": "specializes",
      "target": "EQ6A3ZN48K"
    },
    {
      "field": "defines",
      "target": "4JWHKNGS4Y"
    }
  ],
  "body": "\n## Summary\n\nA character sum is a sum of the values of a [[Dirichlet character|7XB5VJKDPG]] over a range of integers, such as $\\sum_{M \u003c n \\le M + N} \\chi(n)$.\nOver a full period the sum is $0$ for a nonprincipal character.\nSo the interest lies in incomplete sums and in sums of $\\chi(f(n))$ for polynomials $f$.\nBounds for character sums control the distribution of quadratic residues and of primitive roots, and they enter the study of [[Dirichlet L-functions|9D2Z543EJQ]].\n\n## Explanations\n\n### L1 human\n\nA Dirichlet character assigns to each integer a root of unity or $0$, following a periodic and multiplicative pattern.\nFor the Legendre symbol modulo a prime $p$, the value is $+1$ on quadratic residues and $-1$ on non-residues.\nA character sum adds these values over an interval, and so counts residues minus non-residues there.\nA small sum means that residues and non-residues are well mixed in the interval.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "11L40",
      "11L10"
    ],
    "name": "Character sum",
    "same_as": {
      "wikidata": "Q5073749",
      "wikipedia": "Character sum"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Character_sum"
      }
    ],
    "tag": "VTYJ9419C7"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/VTYJ9419C7/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/VTYJ9419C7.md",
  "msc": [
    "11L40",
    "11L10"
  ],
  "name": "Character sum",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/VTY/VTYJ9419C7.md",
  "status": null,
  "tag": "VTYJ9419C7",
  "uri": "m2://VTYJ9419C7",
  "url": "https://math2.tamnd.com/wiki/character-sum/",
  "writing": {
    "tag": "VTYJ9419C7",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}