{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $q \\ge 1$ be an integer.\nA *Dirichlet character modulo $q$* is a function $\\chi \\colon \\Z \\to \\C$ such that:\n\n1. $\\chi(n + q) = \\chi(n)$ for all $n \\in \\Z$.\n2. $\\chi(n) = 0$ if $\\gcd(n, q) \u003e 1$, and $\\chi(n) \\ne 0$ if $\\gcd(n, q) = 1$.\n3. $\\chi(mn) = \\chi(m)\\chi(n)$ for all $m, n \\in \\Z$.\n\nThe *principal character* $\\chi_0$ modulo $q$ has $\\chi_0(n) = 1$ for $n$ [[coprime|0VZ48N083F]] to $q$ and $\\chi_0(n) = 0$ otherwise.\n\n## Remarks\n\nExample: for $q = 4$, let $\\chi(n) = 0$ for even $n$, $\\chi(n) = 1$ for $n \\equiv 1$ and $\\chi(n) = -1$ for $n \\equiv 3 \\pmod 4$.\nClauses 1 and 2 are clear.\nFor clause 3 it suffices to check odd classes: $1 \\cdot 1 \\equiv 1$, $1 \\cdot 3 \\equiv 3$ and $3 \\cdot 3 \\equiv 1 \\pmod 4$ match $1 \\cdot 1 = 1$, $1 \\cdot (-1) = -1$ and $(-1)(-1) = 1$.\nNon-example: for $q = 3$, the function $f(n) = r$, where $r \\in \\set{0, 1, 2}$ is the remainder of $n$ modulo $3$, satisfies clauses 1 and 2 but not clause 3, since $f(2 \\cdot 2) = f(4) = 1 \\ne 4 = f(2) f(2)$.\nEquivalently, a Dirichlet character modulo $q$ is a [[group homomorphism|0JYTG5K9Y4]] $(\\Z/q\\Z)^\\times \\to \\C^\\times$ from the unit group of the [[integers modulo q|03VFZ2WETT]], extended by $0$ on the classes not coprime to $q$.\nHence each value $\\chi(n)$ with $\\gcd(n, q) = 1$ is a $\\varphi(q)$-th root of unity, and there are exactly $\\varphi(q)$ characters modulo $q$, where $\\varphi$ is [[Euler's totient function|0DAMHCT17R]].\nEach Dirichlet character is a [[completely multiplicative function|0RXKW044R4]].\nThe Legendre symbol $n \\mapsto (n/p)$ is a Dirichlet character modulo an odd prime $p$.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "defines": "7XB5VJKDPG",
    "kind": "definition",
    "msc": [
      "11A25",
      "11M06"
    ],
    "name": "Dirichlet character",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Dirichlet_character"
      }
    ],
    "tag": "JR4QSEFXWF",
    "uses": [
      "0VZ48N083F",
      "0RXKW044R4",
      "03VFZ2WETT",
      "0JYTG5K9Y4"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/JR4QSEFXWF/",
  "links": [
    {
      "field": "defines",
      "target": "7XB5VJKDPG"
    },
    {
      "field": "uses",
      "target": "0VZ48N083F"
    },
    {
      "field": "uses",
      "target": "0RXKW044R4"
    },
    {
      "field": "uses",
      "target": "03VFZ2WETT"
    },
    {
      "field": "uses",
      "target": "0JYTG5K9Y4"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/JR4QSEFXWF.md",
  "msc": [
    "11A25",
    "11M06"
  ],
  "name": "Dirichlet character",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/JR4/JR4QSEFXWF.md",
  "status": null,
  "tag": "JR4QSEFXWF",
  "uri": "m2://JR4QSEFXWF",
  "url": "https://math2.tamnd.com/wiki/dirichlet-character-definition/",
  "writing": {
    "tag": "JR4QSEFXWF",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}