Mobius inversion on finite divisor sets

Mathematical statement

For positive integer n, F(n)=sum over d|n of f(d) implies f(n)=sum over d|n of mu(d)*F(n/d). The equivalent indexing mu(n/d)*F(d) is valid but must not be mixed with the first convention. Enumerate every positive divisor and derive mu from complete prime multiplicities. Retain n=1 and mu(1)=1. This is convolution on the divisor poset, not a sum over all d≤n.

Representative formula (exact TeX)

f(n)=I(F,n)

Assumptions

• n is a positive safe integer and divisor enumeration is complete.
• Question families use the exact integer arithmetic function f(k)=a*k+b.

Symbol scope

• f — original arithmetic function
• n — positive integer
• I — Mobius inversion operator
• F — divisor-sum transform

Intuition, definition, and misconceptions

The forward transform uses all divisors.

Keep d and n/d consistently aligned in the inverse.

The roundtrip includes n=1.

Prime/proper divisors alone are insufficient; d|n does not mean d≤n.

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

• Mobius function — Use the mu weights.
• Natural-number divisibility — Enumerate the complete divisor set.
• Basic counting principles — Regroup the finite sums.

Verification boundary

Zero, negative inputs and incomplete factorization/divisor lists are outside the positive domain of this slice.

not_formalized

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

Next obligation

Add the Legendre symbol and Euler criterion.

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.