ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 2026-08-31

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.

Euler's totient as the convolution $μ * {id}_{1}$

Example

The published divisor-sum identity for Euler's totient implies

φ = μ * {id}_{1} .

Equivalently, for every positive integer $n$ ,

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

Facts & Assumptions

Given: A positive integer $n$ .

Verification

technique · direct

1.1

By For every positive integer $n$ , $\sum_{d ∣ n, d > 0} φ (d) = n$ , one has $\sum_{d ∣ n} φ (d) = n = {id}_{1} (n)$ , where ${id}_{1}$ is the power function of The power functions ${id}_{k}$ and the divisor-power-sum functions $σ_{k}$ .

given

2.1

Applying Classical Möbius inversion over positive divisors to the functions $f = φ$ and $g = {id}_{1}$ gives $φ (n) = \sum_{d ∣ n} μ (d) {id}_{1} (n / d) = \sum_{d ∣ n} μ (d) (n / d)$ .

step 1.1

3.1

The final sum is exactly the Dirichlet convolution formula for $μ * {id}_{1}$ , so $φ = μ * {id}_{1}$ .

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

20 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

Karl-Dieter Crisman, Number Theory: In Context and Interactive, Section 23.3 (standard reference, not scraped)

Euler's totient as the convolution μ∗id⁡1μ∗id1​