{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "16SRNY6AXK"
    }
  ],
  "body": "\n## Summary\n\nA Lambert series is a series of the form $\\sum_{n \\ge 1} a_n \\frac{q^n}{1 - q^n}$ in a variable $q$ with $\\abs{q} \u003c 1$.\nExpanding each fraction as a geometric series turns it into a power series whose coefficients are divisor sums $\\sum_{d \\mid m} a_d$.\nThis makes Lambert series a bridge between [[arithmetic functions|09115NQZ07]] and power series, used in the theory of partitions, modular forms and sums of squares.\n\n## Explanations\n\n### L1 human\n\nEach term $q^n / (1 - q^n)$ is the geometric series $q^n + q^{2n} + q^{3n} + \\cdots$, which collects the powers of $q$ whose exponent is a multiple of $n$.\nAdding these up with weights $a_n$, the power $q^m$ receives the weight $a_d$ once for each divisor $d$ of $m$.\nFor example, with all $a_n = 1$, the coefficient of $q^m$ is the number of divisors of $m$, so $\\sum_n q^n/(1 - q^n) = q + 2q^2 + 2q^3 + 3q^4 + \\cdots$.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "kind": "concept",
    "msc": [
      "30B50",
      "11A25"
    ],
    "name": "Lambert series",
    "same_as": {
      "wikidata": "Q1577138",
      "wikipedia": "Lambert series"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Lambert_series"
      }
    ],
    "tag": "HATVWNZNMR"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/HATVWNZNMR/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/HATVWNZNMR.md",
  "msc": [
    "30B50",
    "11A25"
  ],
  "name": "Lambert series",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/HAT/HATVWNZNMR.md",
  "status": null,
  "tag": "HATVWNZNMR",
  "uri": "m2://HATVWNZNMR",
  "url": "https://math2.tamnd.com/wiki/lambert-series/",
  "writing": {
    "tag": "HATVWNZNMR",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}