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

Statement

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

$\mu(mn)=\mu(m)\mu(n).$

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 $mn$ has a unique product form $ab$ with $a\mid m$ and $b\mid n$ (The fundamental theorem of arithmetic: every integer $n \ge 1$ 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 $q_i = p_{\pi(i)}$ for some $\pi \in \operatorname{Sym}(r)$ ).

[L2]

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

[L3]

$\mu(k)=\mu_{\mid}(1,k)$ for every positive integer $k$ (The number-theoretic Möbius function is the poset Möbius function of divisibility: $\mu(n)=\mu_{\mid}(1,n)$ ).

[L4]

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

Proof

technique · direct

1.1

By [L1], $(a,b)\mapsto ab$ is a bijection from $[1,m]_{\mid}\times[1,n]_{\mid}$ to $[1,mn]_{\mid}$ . 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]: $\mu_{\mid}(1,mn)=\mu_{\mid}(1,m)\mu_{\mid}(1,n)$ .

step 1.1L2L4

3.1

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

step 2.1L3∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 104 results over 26 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

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