Mobius function

Mathematical statement

A complete unique-prime-factorization certificate supplies multiplicities and the distinct-prime count, independently of factor order. The exact divisor-sum cross-check is sum over d dividing n of mu(d)=1 for n=1 and0 otherwise. Zero and negative integers are outside the domain. Incomplete trial division does not justify classification.

Representative formula (exact TeX)

mu(n)

Assumptions

• The input is a positive integer and its prime factorization is complete.
• Bounded companions trial-divide completely through the square root of the remaining factor.

Symbol scope

• mu — Mobius function
• n — positive integer

Intuition, definition, and misconceptions

mu(1)=1.

A repeated prime factor gives0.

For a squarefree integer with k distinct prime factors the sign is (−1)^k.

Counting distinct primes while ignoring square multiplicities is insufficient; zero is not classified by this function.

Adopted derivation guide

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Prime natural number — Every factor must be prime.
• Unique prime factorization of integers — Prime multiplicities are required.
• Function mapping — This is an arithmetic function.

Verification boundary

Do not output a Mobius value from an incomplete factorization.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.

Next obligation

Add Mobius inversion.

Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.