CorollaryStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-07-31

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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.

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The number-theoretic Möbius function is multiplicative on coprime positive integers

Statement

If $m, n$ are coprime positive integers, then

$μ (m n) = μ (m) μ (n) .$

Facts & Assumptions

Given: Coprime positive integers $m, n$ (Coprime integers: $\gcd (a, b) = 1$ ).

[L1]

Unique prime factorisation implies that coprime positive integers have disjoint prime supports, and every divisor of $m n$ has a unique product form $a b$ with $a ∣ m$ and $b ∣ n$ (The fundamental theorem of arithmetic: every integer $n \geq 1$ is a product of primes, and the factorisation is unique up to order — if $\prod_{i < r} p_{i} = \prod_{j < s} q_{j}$ with every $p_{i}$ and $q_{j}$ prime, then $r = s$ and $q_{i} = p_{π (i)}$ for some $π \in Sym (r)$ ).

[L2]

Möbius functions multiply on product posets (The Möbius function of a product poset is the product of the Möbius functions).

[L3]

$μ (k) = μ_{∣} (1, k)$ for every positive integer $k$ (The number-theoretic Möbius function is the poset Möbius function of divisibility: $μ (n) = μ_{∣} (1, n)$ ).

[L4]

Möbius values transport through poset isomorphisms because the diagonal and recurrence conditions uniquely determine them (The Möbius recurrence: $μ_{P} (x, x) = 1$ and both interval sums of $μ_{P}$ vanish when $x < y$ ).

Proof

technique · direct

1.1

By [L1], $(a, b) \mapsto a b$ is a bijection from $[1, m]_{∣} \times [1, n]_{∣}$ to $[1, m n]_{∣}$ . It preserves and reflects divisibility coordinatewise, again by the disjoint prime supports, so it is a poset isomorphism.

L1construct

2.1

Apply the product theorem at the endpoints and transfer along the isomorphism by [L4]: $μ_{∣} (1, m n) = μ_{∣} (1, m) μ_{∣} (1, n)$ .

step 1.1L2L4

3.1

Replacing each poset value in step 2.1 by its number-theoretic value using [L3] yields $μ (m n) = μ (m) μ (n)$ .

step 2.1L3∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

44 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

Stanford Pairing-Based Cryptography notes, Möbius inversion (standard reference, not scraped)