Möbius Inversion

Recovering a Function from Divisor Sums

Many arithmetic functions are defined through sums over divisors. For example,

and

A natural question arises: if a function $F(n)$ is defined by summing another function $f(n)$ over the divisors of $n$, can one recover $f$ from $F$?

Möbius inversion gives the answer.

Statement of Möbius Inversion

Suppose $f$ and $F$ are arithmetic functions satisfying

Then

Equivalently,

This formula inverts divisor summation.

Convolution Form

Using Dirichlet convolution, the hypothesis

can be written compactly as

where $\mathbf{1}(n)=1$.

Since

the Möbius function is the convolution inverse of $\mathbf{1}$.

Convolving both sides with $\mu$,

By associativity,

Thus

Written explicitly,

This is Möbius inversion.

First Example

Let

Since

we have

with

Applying Möbius inversion,

Thus the constant function $1$ can be recovered from the divisor-counting function through Möbius inversion.

Recovering Euler’s Totient Function

Earlier we proved

Thus

where

Applying Möbius inversion gives

Therefore

For example, if

then the divisors are

Only squarefree divisors contribute:

This matches the direct calculation.

Inclusion-Exclusion Interpretation

Möbius inversion is an arithmetic form of inclusion-exclusion.

The Möbius function alternates signs according to the number of prime factors:

for squarefree integers with $r$ distinct prime factors.

Thus Möbius inversion systematically adds and subtracts contributions from divisibility conditions.

For example, suppose we wish to count integers up to $x$ that are coprime to $n$. We remove multiples of primes dividing $n$, add back multiples of products of two such primes, subtract multiples of products of three such primes, and so on.

This alternating process is encoded exactly by the Möbius function.

Counting Primitive Objects

Möbius inversion often separates primitive objects from nonprimitive ones.

Suppose $A(n)$ counts all objects of size $n$, including repetitions arising from smaller periods, while $P(n)$ counts primitive objects. Frequently one has a relation

Then Möbius inversion gives

This principle appears in combinatorics, dynamical systems, and finite field theory.

Multiplicative Form

If $f$ and $F$ are multiplicative and

then Möbius inversion preserves multiplicativity.

Since $\mu$ is multiplicative and convolution of multiplicative functions is multiplicative,

is multiplicative whenever $F$ is.

Thus Möbius inversion is compatible with prime factorization.

Generalized Form

More generally, suppose $g$ is an arithmetic function with Dirichlet inverse $g^{-1}$. If

then

Möbius inversion is the special case

Thus Möbius inversion is part of a larger algebraic theory of arithmetic functions under Dirichlet convolution.

Dirichlet Series Interpretation

Dirichlet convolution corresponds to multiplication of Dirichlet series. Since

their Dirichlet series satisfy

But

Therefore

This analytic identity reflects the algebraic fact that $\mu$ is the inverse of $\mathbf{1}$ under convolution.

Role in Number Theory

Möbius inversion is one of the central tools of multiplicative number theory. It reverses divisor summation and transforms global divisor information into local arithmetic data.

It appears in counting problems, coprimality arguments, inclusion-exclusion formulas, analytic number theory, finite fields, and combinatorics.

The key principle is that divisor sums are invertible because the Möbius function provides the arithmetic analogue of alternating cancellation.