# Mobius function

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

On positive integers, complete prime multiplicities determine the Mobius function: mu(1)=1, a repeated prime gives0, and a squarefree integer has the parity sign.

mko-mobius-function · definition · v0.10.0



## 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

[Complete factorization and mu(30)](https://ocme.evemisslab.com/en/derivations/derivation-mko-mobius-function-01/)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](https://ocme.evemisslab.com/en/mko/mko-prime-number/) — Every factor must be prime.
- [Unique prime factorization of integers](https://ocme.evemisslab.com/en/mko/mko-unique-prime-factorization/) — Prime multiplicities are required.
- [Function mapping](https://ocme.evemisslab.com/en/mko/mko-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.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/mobius-function.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-mobius-function/index.md)