{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "24GAZHVTGB"
    },
    {
      "field": "uses",
      "target": "63FS7T8R18"
    },
    {
      "field": "uses",
      "target": "6ZGF49H346"
    },
    {
      "field": "uses",
      "target": "TXB2F07CC6"
    },
    {
      "field": "uses",
      "target": "YQ21C2N2HX"
    },
    {
      "field": "specializes",
      "target": "9D2Z543EJQ"
    },
    {
      "field": "specializes",
      "target": "KXS5DQDFYD"
    },
    {
      "field": "defines",
      "target": "HN0AXS6VSG"
    }
  ],
  "body": "\n## Summary\n\nA Dirichlet series is a series $\\sum_{n \\ge 1} a(n) n^{-s}$, where $a$ is an arithmetic function and $s$ is a complex variable.\nIt converges in a right half-plane and defines a holomorphic function there.\nMultiplying two Dirichlet series corresponds to the [[Dirichlet convolution|037F589D1D]] of their coefficients, which makes them the natural generating functions of multiplicative number theory.\nThe [[Riemann zeta function|1XWBY4N0VT]] and the [[Dirichlet L-functions|9D2Z543EJQ]] are the main examples.\n\n## Explanations\n\n### L1 human\n\nA power series $\\sum a_n z^n$ packs a sequence into a function in a way suited to adding indices.\nA Dirichlet series $\\sum a(n) n^{-s}$ does the same in a way suited to multiplying indices, since $m^{-s} n^{-s} = (mn)^{-s}$.\nFor the constant sequence $a(n) = 1$ one obtains $\\zeta(s) = 1 + 2^{-s} + 3^{-s} + \\cdots$.\nFacts about divisors and primes then become facts about products and quotients of such functions.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "11M41",
      "30B50"
    ],
    "name": "Dirichlet series",
    "same_as": {
      "wikidata": "Q620595",
      "wikipedia": "Dirichlet series"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Dirichlet_series"
      }
    ],
    "tag": "8MWTEF6462"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/8MWTEF6462/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/8MWTEF6462.md",
  "msc": [
    "11M41",
    "30B50"
  ],
  "name": "Dirichlet series",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/8MW/8MWTEF6462.md",
  "status": null,
  "tag": "8MWTEF6462",
  "uri": "m2://8MWTEF6462",
  "url": "https://math2.tamnd.com/wiki/dirichlet-series/",
  "writing": {
    "tag": "8MWTEF6462",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}