{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "YSMBT6KQ4N"
    }
  ],
  "body": "\n## Summary\n\nLambert summation is a [[summability method|8X25Q4XYMN]] built on [[Lambert series|HATVWNZNMR]] $\\sum n a_n \\frac{x^n}{1 - x^n}$, multiplied by $1 - x$, with $x \\to 1$ from below.\nIt is regular, so it gives the usual sum to every convergent series.\nIt is closely tied to the [[prime number theorem|0EHBY5Y84N]].\nThe series $\\sum_{n \\ge 1} \\mu(n)/n$ is Lambert summable to $0$ by a short computation, and a Tauberian theorem upgrades this to convergence.\n\n## Explanations\n\n### L1 human\n\nA Lambert series groups the terms of a series along divisors: $\\frac{x^n}{1 - x^n} = x^n + x^{2n} + x^{3n} + \\cdots$.\nLambert summation uses this grouping as a damping device.\nIt suits series from number theory, whose terms are built from divisors and primes.\nFor the Möbius function, the grouping makes almost everything cancel, which gives a quick Lambert sum.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "40G10",
      "11M45"
    ],
    "name": "Lambert summation",
    "same_as": {
      "wikidata": "Q6481320",
      "wikipedia": "Lambert summation"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Lambert_summation"
      }
    ],
    "specializes": [
      "8X25Q4XYMN"
    ],
    "tag": "W8H1WX0JHG"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/W8H1WX0JHG/",
  "links": [
    {
      "field": "specializes",
      "target": "8X25Q4XYMN"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/W8H1WX0JHG.md",
  "msc": [
    "40G10",
    "11M45"
  ],
  "name": "Lambert summation",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/W8H/W8H1WX0JHG.md",
  "status": null,
  "tag": "W8H1WX0JHG",
  "uri": "m2://W8H1WX0JHG",
  "url": "https://math2.tamnd.com/wiki/lambert-summation/",
  "writing": {
    "tag": "W8H1WX0JHG",
    "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
      }
    ]
  }
}