{
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  "category": "Number theory",
  "claimBoundary": "This page indexes Mathlib's additive-commutative-group form of Möbius inversion. For functions f, g : ℕ → R and every positive n, it identifies the divisor-sum relation g(n) = ∑_{d ∣ n} f(d) with recovery of f(n) by the Möbius-weighted sum over ordered factor pairs in n.divisorsAntidiagonal. The integer-valued Möbius function acts on R by scalar multiplication. The declaration excludes n = 0 and is not every informal inversion formulation.",
  "collectionSlug": "landmark-theorems-in-mathlib",
  "exactFormalStatement": "theorem ArithmeticFunction.sum_eq_iff_sum_smul_moebius_eq {R : Type*} [AddCommGroup R] {f g : ℕ → R} : (∀ n > 0, ∑ i ∈ n.divisors, f i = g n) ↔ ∀ n > 0, ∑ x ∈ n.divisorsAntidiagonal, μ x.fst • g x.snd = f n",
  "family": "Arithmetic functions and inversion",
  "id": "library.mathlib.mobius-inversion.v001",
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    "completeFormalType": "theorem ArithmeticFunction.sum_eq_iff_sum_smul_moebius_eq {R : Type*} [AddCommGroup R] {f g : ℕ → R} : (∀ n > 0, ∑ i ∈ n.divisors, f i = g n) ↔ ∀ n > 0, ∑ x ∈ n.divisorsAntidiagonal, μ x.fst • g x.snd = f n",
    "concepts": [
      "Möbius inversion",
      "arithmetic functions",
      "divisor sums",
      "Dirichlet convolution",
      "zeta inverse"
    ],
    "plainLanguage": "Suppose g records the sum of f over every positive divisor of n. Then f can be recovered by summing the values of g along the factor pairs of n, weighted by the integer-valued Möbius function—and this recovery relation is equivalent to the original divisor-sum relation.",
    "poster": {
      "alt": "An ivory and forest-green theorem poster displays the positive-index divisor-sum equivalence, a compact divisor lattice, and the checked route through the arithmetic zeta function and its Möbius inverse.",
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      "caption": "For additive-commutative-group-valued functions on positive naturals, Möbius weighting exactly recovers a function from its divisor sums.",
      "description": "The poster presents the selected equivalence without flattening its ordered-factor-pair recovery sum. A sparse central engraving moves from divisor aggregation to signed cancellation, while a short route names the zeta action, Möbius action, and two-sided inverse identity used in the Mathlib proof.",
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      "generatedAt": "2026-07-25T20:39:31Z",
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      "publicPath": "/assets/library-theorems/mobius-inversion/theorem-poster-v1.png",
      "sha256": "sha256:faeef6644141b3f905678487fa27004e25cb503f0847197324dbb3c61399d4d1",
      "title": "Möbius Inversion — divisor sums reversed",
      "width": 1024,
      "kind": "theorem_poster",
      "transcript": "NUMBER THEORY · ARITHMETIC FUNCTIONS\nMÖBIUS INVERSION\nFor every n > 0\n\ng(n) = ∑ d ∈ n.divisors, f(d)\n⇔\nf(n) = ∑ (a,b) ∈ n.divisorsAntidiagonal, μ(a) • g(b)\n\nDIVISOR SUM · MÖBIUS RECOVERY\n\nTHE CHECKED ROUTE\nζ • f = g\nμ • g = f\nμ ∗ ζ = ζ ∗ μ = 1\n\nEXACT SCOPE\nAdditive commutative groups. Positive n only.\nμ is integer-valued and acts by scalar multiplication."
    },
    "visual": {
      "alt": "A text-free engraved divisor lattice first gathers values from all positive divisors into one total, then passes through a Möbius-weighted factor-pair fan whose cancellations isolate the original value.",
      "byteSize": 3115060,
      "caption": "The zeta divisor sum gathers arithmetic data; Möbius weighting reverses that accumulation on every positive index.",
      "description": "On a dark forest-green field, ivory divisor nodes flow into an antique-gold aggregate. A mirrored factor-pair fan carries distinct positive, negative, and vanishing Möbius tokens; paired cancellations leave one isolated source node, visualizing the arithmetic zeta function and Möbius function as inverse operators.",
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          "height": 427,
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      "generatedAt": "2026-07-25T20:39:31Z",
      "height": 1024,
      "mediaType": "image/png",
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      "publicPath": "/assets/library-theorems/mobius-inversion/theorem-schematic-v1.png",
      "sha256": "sha256:3b03033693fbb0262e4dfa06ff3bdff10eb2248e191ce597cb80ff3c47b6bc64",
      "title": "Divisor aggregation and Möbius recovery",
      "width": 1536
    },
    "walkthrough": null,
    "whyLandmark": "Möbius inversion is a basic change-of-coordinates principle for arithmetic data: cumulative information over divisors can be recovered exactly through the convolution inverse of the constant-one arithmetic function. It underlies multiplicative number theory, inclusion–exclusion on divisibility, and many transformations of arithmetic functions."
  },
  "nonClaims": [
    "The selected statement ranges only over positive natural n; it does not assert either relation at n = 0.",
    "The codomain is an additive commutative group, and the integer-valued Möbius function acts by scalar multiplication.",
    "The recovery sum uses n.divisorsAntidiagonal, whose entries are ordered factor pairs; it is not written as a plain sum over an unspecified index.",
    "This declaration is one precise additive form of Möbius inversion, not a claim for every informal inversion formula.",
    "ProofAtlas is indexing an existing Mathlib theorem, not claiming new mathematics."
  ],
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  "statementId": "statement.library.mathlib.mobius-inversion.v001",
  "statusAxes": {
    "acceptedAtlasResult": "Not recorded for the preferred artifact",
    "locallyReproduced": "Exact upstream declaration replayed",
    "reviewedPage": "Current public presentation reviewed",
    "upstreamIndexed": "Pinned source bytes verified locally"
  },
  "summary": "Divisor summation by the arithmetic zeta function is inverted by Möbius weighting on ordered factor pairs.",
  "targetId": "target.library.mathlib.mobius-inversion.v001",
  "title": "Möbius Inversion",
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