{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $A \\colon [0, \\infty) \\to [0, \\infty)$ be a nondecreasing [[monotone function|0WFF6AZPBP]].\nAssume that the [[Laplace transform|05XT5AVJBK]]\n$$f(s) = \\int_0^\\infty A(x) e^{-xs} \\, dx$$\nconverges for every complex $s$ with $\\operatorname{Re} s \u003e 1$.\nAssume also that there is a real number $c \\ge 0$ such that $f(s) - \\frac{c}{s - 1}$ extends to a continuous function on the closed half-plane $\\operatorname{Re} s \\ge 1$.\nThen\n$$\\lim_{x \\to \\infty} e^{-x} A(x) = c.$$\n\n## Remarks\n\nThe proof is Fourier-analytic.\nOne integrates $f(s) - \\frac{c}{s-1}$ on vertical lines against a Fejér-type kernel with compactly supported Fourier transform, lets $\\operatorname{Re} s \\to 1^+$, and uses the Riemann–Lebesgue lemma; monotonicity of $A$ then removes the smoothing.\n\nExample: the [[prime number theorem|0EHBY5Y84N]].\nLet $\\psi(y) = \\sum_{n \\le y} \\Lambda(n)$ with the [[von Mangoldt function|087EMZTHWN]] $\\Lambda$, and put $A(x) = \\psi(e^x)$.\nThen $\\int_0^\\infty \\psi(e^x) e^{-xs} \\, dx = -\\frac{\\zeta'(s)}{s \\, \\zeta(s)}$ for $\\operatorname{Re} s \u003e 1$.\nThis differs from $\\frac{1}{s - 1}$ by a function continuous on $\\operatorname{Re} s \\ge 1$, because $\\zeta$ has a simple pole at $1$ and [[does not vanish on the line|CXES7915R7]] $\\operatorname{Re} s = 1$.\nSo $\\psi(y)/y \\to 1$, which is equivalent to the prime number theorem.\n\nThe hypothesis is only continuity up to the boundary line, not analytic continuation beyond it.\n\n## Explanations\n\n### L1 human\n\nThe theorem reads the growth of a nondecreasing function off its Laplace transform.\nIf the transform behaves like $\\frac{c}{s-1}$ near the line $\\operatorname{Re} s = 1$ and has no other trouble on that line, then the function grows like $c e^x$.\nApplied to the primes, it turns the fact that the zeta function has no zeros on the line $\\operatorname{Re} s = 1$ into the prime number theorem.\n\n### L2 human\n\nIn Dirichlet series form: if $a_n \\ge 0$ and $F(s) = \\sum a_n n^{-s}$ converges for $\\operatorname{Re} s \u003e 1$ with $F(s) - \\frac{c}{s-1}$ continuous on $\\operatorname{Re} s \\ge 1$, then $\\sum_{n \\le y} a_n \\sim c\\,y$.\nThis follows by applying the theorem to $A(x) = \\sum_{n \\le e^x} a_n$.\nThe theorem is a special case of Wiener's general Tauberian theory; Newman's later contour-integral proof of the prime number theorem uses a closely related Tauberian argument.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "about": [
      "D6XWK1NRY1"
    ],
    "kind": "statement",
    "msc": [
      "11M45",
      "40E05",
      "44A10"
    ],
    "name": "Wiener–Ikehara theorem",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q7999144",
      "wikipedia": "Wiener–Ikehara theorem"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Wiener–Ikehara_theorem"
      }
    ],
    "tag": "KQ57DC3AKM",
    "uses": [
      "D6XWK1NRY1",
      "0WFF6AZPBP",
      "05XT5AVJBK",
      "087EMZTHWN"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/KQ57DC3AKM/",
  "links": [
    {
      "field": "about",
      "target": "D6XWK1NRY1"
    },
    {
      "field": "uses",
      "target": "D6XWK1NRY1"
    },
    {
      "field": "uses",
      "target": "0WFF6AZPBP"
    },
    {
      "field": "uses",
      "target": "05XT5AVJBK"
    },
    {
      "field": "uses",
      "target": "087EMZTHWN"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/KQ57DC3AKM.md",
  "msc": [
    "11M45",
    "40E05",
    "44A10"
  ],
  "name": "Wiener–Ikehara theorem",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/KQ5/KQ57DC3AKM.md",
  "status": null,
  "tag": "KQ57DC3AKM",
  "uri": "m2://KQ57DC3AKM",
  "url": "https://math2.tamnd.com/wiki/wiener-ikehara-theorem/",
  "writing": {
    "tag": "KQ57DC3AKM",
    "kind": "statement",
    "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
      }
    ]
  }
}