{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "J900ENTQ5B"
    }
  ],
  "body": "\n## Summary\n\nA [[Dirichlet series|8MWTEF6462]] $F(s) = \\sum_{n \\ge 1} a(n) n^{-s}$ can be written as a [[Mellin transform|N5PVGRJE1T]] in two standard ways.\nFirst, $\\Gamma(s) F(s)$ is the Mellin transform of the exponential sum $\\sum_n a(n) e^{-nx}$.\nSecond, $F(s)/s$ is the Mellin transform at $-s$ of the summatory function $A(x) = \\sum_{n \\le x} a(n)$.\nThese identities let one study Dirichlet series with the tools of integral transforms: analytic continuation, inversion formulas such as Perron's formula, and asymptotics through residues.\n\n## Explanations\n\n### L1 human\n\nThe basic fact is $\\int_0^\\infty x^{s-1} e^{-nx} \\, dx = \\Gamma(s) n^{-s}$.\nSo a sum of exponentials $\\sum a(n) e^{-nx}$ has a Mellin transform equal to $\\Gamma(s)$ times the Dirichlet series with the same coefficients.\nFor $a(n) = 1$ the exponential sum is $1/(e^x - 1)$, and the identity becomes $\\int_0^\\infty x^{s-1}/(e^x - 1) \\, dx = \\Gamma(s)\\zeta(s)$ for $\\operatorname{Re} s \u003e 1$.\nGoing back by Mellin inversion recovers information on the coefficients from the analytic behaviour of $F$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "11M41",
      "44A15",
      "30B50"
    ],
    "name": "Dirichlet series as Mellin transforms",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Dirichlet_series"
      }
    ],
    "tag": "K5RXH32SE7"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/K5RXH32SE7/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/K5RXH32SE7.md",
  "msc": [
    "11M41",
    "44A15",
    "30B50"
  ],
  "name": "Dirichlet series as Mellin transforms",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/K5R/K5RXH32SE7.md",
  "status": null,
  "tag": "K5RXH32SE7",
  "uri": "m2://K5RXH32SE7",
  "url": "https://tamnd.github.io/math2/wiki/dirichlet-series-as-mellin-transforms/",
  "writing": {
    "tag": "K5RXH32SE7",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}