ExampleConstruction: AI-adaptedVerification: Not suppliedaudited 2026-07-31

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Möbius inversion of $\sum_{d ∣ n} φ (d) = n$ gives $φ (n) = \sum_{d ∣ n} μ (d) (n / d)$

Example

The divisor-sum theorem from the CRT development states

$n = \sum_{d ∣ n} φ (d)$

for every positive integer $n$ (For every positive integer $n$ , $\sum_{d ∣ n, d > 0} φ (d) = n$ ). Apply Classical Möbius inversion over positive divisors with $g (n) = n$ and $f (n) = φ (n)$ . The form indexed by the complementary divisor gives

$φ (n) = \sum_{d ∣ n} μ (d) g (n / d) = \sum_{d ∣ n} μ (d) \frac{n}{d} .$

At $n = 12$ , the values from The number-theoretic Möbius function $μ (n)$ from prime factorisation give

$φ (12) = 12 - 6 - 4 + 0 + 2 + 0 = 4,$

where the terms correspond to $d = 1, 2, 3, 4, 6, 12$ . Thus Möbius inversion recovers $φ (12) = 4$ directly from the CRT divisor-sum theorem.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

24 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

P. J. Cameron, Notes on Number Theory, Theorem 7.11 (standard reference, not scraped)
Stanford Pairing-Based Cryptography notes, Möbius inversion (standard reference, not scraped)

Möbius inversion of ∑d∣nφ(d)=n∑d∣n​φ(d)=n gives φ(n)=∑d∣nμ(d)(n/d)φ(n)=∑d∣n​μ(d)(n/d)