# Möbius inversion

###### Theorem 1 (Moebius Inversion).

Let $\mu$ be the Moebius function, and let $f$ and $g$ be two functions on the positive integers. Then the following two conditions are equivalent     :

 $\displaystyle f(n)$ $\displaystyle=$ $\displaystyle\sum_{d|n}g(d)\textrm{ for all }n\in\mathbb{N}^{*}$ (1) $\displaystyle g(n)$ $\displaystyle=$ $\displaystyle\sum_{d|n}\mu(d)f\left(\frac{n}{d}\right)\textrm{ for all }n\in% \mathbb{N}^{*}$ (2)

Proof: Fix some $n\in\mathbb{N}^{*}$. Assuming (1), we have

 $\displaystyle\sum_{d|n}\mu(d)f\left(\frac{n}{d}\right)$ $\displaystyle=$ $\displaystyle\sum_{d|n}\mu(d)\sum_{e|n/d}g(e)$ $\displaystyle=$ $\displaystyle\sum_{k|n}\sum_{d|k}\mu(d)g\left(\frac{n}{k}\right)$ $\displaystyle=$ $\displaystyle\sum_{k|n}g\left(\frac{n}{k}\right)\sum_{d|k}\mu(d)$ $\displaystyle=$ $\displaystyle g(n),$

Conversely, assuming (2), we get

 $\displaystyle\sum_{d|n}g(d)$ $\displaystyle=$ $\displaystyle\sum_{d|n}\sum_{e|d}\mu(e)f\left(\frac{d}{e}\right)$ $\displaystyle=$ $\displaystyle\sum_{k|n}f\left(\frac{n}{k}\right)\sum_{e|k}\mu\left(\frac{k}{e}\right)$ $\displaystyle=$ $\displaystyle f(n)\qquad\text{(by the same identity)}$

as claimed.

## Möbius-Rota inversion

G.-C. Rota has described a generalization  of the Möbius formalism. In it, the set $\mathbb{N}^{*}$, ordered by the relation    $x|y$ between elements $x$ and $y$, is replaced by a more general ordered set, and $\mu$ is replaced by a function of two variables.

Let $(S,\leq)$ be a locally finite  ordered set, i.e. an ordered set such that $\{z\in S|x\leq z\leq y\}$ is a finite set  for all $x,y\in S$. Let $A$ be the set of functions $\alpha:S\times S\to\mathbb{Z}$ such that

 $\displaystyle\alpha(x,x)$ $\displaystyle=$ $\displaystyle 1\textrm{ for all }x\in S$ (3) $\displaystyle\alpha(x,y)$ $\displaystyle\neq$ $\displaystyle 0\textrm{ implies }x\leq y$ (4)

$A$ becomes a monoid if we define the product   of any two of its elements, say $\alpha$ and $\beta$, by

 $(\alpha\beta)(x,y)=\sum_{t\in S}\alpha(x,t)\beta(t,y).$

The sum makes sense because $\alpha(x,t)\beta(t,y)$ is nonzero for only finitely many values of $t$. (Clearly this definition is akin to the definition of the product of two square matrices.)

Consider the element $\iota$ of $A$ defined simply by

 $\iota(x,y)=\begin{cases}1&\textrm{ if x\leq y}\\ 0&\text{otherwise.}\end{cases}$

The function $\iota$, regarded as a matrix over $\mathbb{Z}$, has an inverse matrix, say $\nu$. That means

 $\sum_{t\in S}\iota(x,t)\nu(t,y)=\begin{cases}1&\text{if x=y,}\\ 0&\text{otherwise.}\end{cases}$

Thus for any $f,g\in A$, the equations

 $\displaystyle f$ $\displaystyle=$ $\displaystyle\iota g$ (5) $\displaystyle g$ $\displaystyle=$ $\displaystyle\nu f$ (6)

are equivalent.

Now let’s sketch out how the traditional Möbius inversion is a special case of Rota’s notion. Let $S$ be the set $\mathbb{N}^{*}$, ordered by the relation $x|y$ between elements $x$ and $y$. In this case, $\nu$ is essentially a function of only one variable:

With the above notation, $\nu(x,y)=\mu(y/x)$ for all $x,y\in\mathbb{N}^{*}$ such that $x|y$.

The proof is fairly straightforward, by induction  on the number of elements of the interval $\{z\in S|x\leq z\leq y\}$.

Now let $g$ be a function from $\mathbb{N}^{*}$ to some additive group  , and write $\overline{g}(x,y)=g(y/x)$ for all pairs $(x,y)$ such that $x|y$.

Example: Let $E$ be a set, and let $S$ be the set of all finite subsets of $E$, ordered by inclusion. The ordered set $S$ is left-finite, and for any $x,y\in S$ such that $x\subset y$, we have $\nu(x,y)=(-1)^{|y-x|}$, where $|z|$ denotes the cardinal of the finite set $z$.

## An Additional Generalization

A final generalization of Moebius inversion occurs when the sum is taken over all integers less than some real value $x$ rather than over the divisors   of an integer. Specifically, let $f:\mathbb{R}\rightarrow\mathbb{C}$ and define $F:\mathbb{R}\rightarrow\mathbb{C}$ by $F(x)=\sum_{n\leq x}f(\frac{x}{n})$.

Then

 $\displaystyle f(x)=\sum_{n\leq x}\mu(n)F\left(\frac{x}{n}\right).$
 Title Möbius inversion Canonical name MobiusInversion Date of creation 2013-03-22 11:46:58 Last modified on 2013-03-22 11:46:58 Owner mathcam (2727) Last modified by mathcam (2727) Numerical id 25 Author mathcam (2727) Entry type Topic Classification msc 11A25 Synonym Moebius inversion Related topic MoebiusFunction Defines Mobius inversion formula Defines Mobius transform Defines Mobius function Defines Mobius-Rota inversion