Möbius inversion formula

Template:Short description Template:Redirect-distinguish In mathematics, the classic Möbius inversion formula is a relation between pairs of arithmetic functions, each defined from the other by sums over divisors. It was introduced into number theory in 1832 by August Ferdinand Möbius.<ref>Template:Harvnb</ref>

A large generalization of this formula applies to summation over an arbitrary locally finite partially ordered set, with Möbius' classical formula applying to the set of the natural numbers ordered by divisibility: see incidence algebra.

Statement of the formulaEdit

The classic version states that if Template:Mvar and Template:Mvar are arithmetic functions satisfying

<math>g(n)=\sum_{d \mid n}f(d)\quad\text{for every integer }n\ge 1</math>

then

<math>f(n)=\sum_{d \mid n}\mu(d)\,g\!\left(\frac{n}{d}\right)\quad\text{for every integer }n\ge 1</math>

where Template:Mvar is the Möbius function and the sums extend over all positive divisors Template:Mvar of Template:Mvar (indicated by <math>d \mid n</math> in the above formulae). In effect, the original Template:Math can be determined given Template:Math by using the inversion formula. The two sequences are said to be Möbius transforms of each other.

The formula is also correct if Template:Mvar and Template:Mvar are functions from the positive integers into some abelian group (viewed as a Template:Math-module).

In the language of Dirichlet convolutions, the first formula may be written as

<math>g=\mathit{1}*f</math>

where Template:Math denotes the Dirichlet convolution, and Template:Math is the constant function Template:Math. The second formula is then written as

Many specific examples are given in the article on multiplicative functions.

The theorem follows because Template:Math is (commutative and) associative, and Template:Math, where Template:Mvar is the identity function for the Dirichlet convolution, taking values Template:Math, Template:Math for all Template:Math. Thus

<math>\mu * g = \mu * (\mathit{1} * f) = (\mu * \mathit{1}) * f = \varepsilon * f = f</math>.

Replacing <math>f, g</math> by <math>\ln f, \ln g</math>, we obtain the product version of the Möbius inversion formula:

<math>g(n) = \prod_{d|n} f(d) \iff f(n) = \prod_{d|n} g\left(\frac{n}{d}\right)^{\mu(d)}, \forall n \geq 1.</math>

Series relationsEdit

Let

so that

<math>b_n=\sum_{d\mid n} \mu\left(\frac{n}{d}\right)a_d</math>

is its transform. The transforms are related by means of series: the Lambert series

<math>\sum_{n=1}^\infty a_n x^n = \sum_{n=1}^\infty b_n \frac{x^n}{1-x^n}</math>

and the Dirichlet series:

<math>\sum_{n=1}^\infty \frac {a_n} {n^s} = \zeta(s)\sum_{n=1}^\infty \frac{b_n}{n^s}</math>

where Template:Math is the Riemann zeta function.

Repeated transformationsEdit

Given an arithmetic function, one can generate a bi-infinite sequence of other arithmetic functions by repeatedly applying the first summation.

For example, if one starts with Euler's totient function Template:Mvar, and repeatedly applies the transformation process, one obtains:

Template:Mvar the totient function
Template:Math, where Template:Math is the identity function
Template:Math, the divisor function

If the starting function is the Möbius function itself, the list of functions is:

Template:Mvar, the Möbius function
Template:Math where <math display="block">\varepsilon(n) = \begin{cases} 1, & \text{if }n=1 \\ 0, & \text{if }n>1 \end{cases} </math> is the unit function
Template:Math, the constant function
Template:Math, where Template:Math is the number of divisors of Template:Mvar, (see divisor function).

Both of these lists of functions extend infinitely in both directions. The Möbius inversion formula enables these lists to be traversed backwards.

As an example the sequence starting with Template:Mvar is:

<math>f_n =

 \begin{cases}
  \underbrace{\mu * \ldots * \mu}_{-n \text{ factors}} * \varphi & \text{if } n < 0 \\[8px]
  \varphi & \text{if } n = 0 \\[8px]
  \varphi * \underbrace{\mathit{1}* \ldots * \mathit{1}}_{n \text{ factors}} & \text{if } n > 0
 \end{cases}

</math>

The generated sequences can perhaps be more easily understood by considering the corresponding Dirichlet series: each repeated application of the transform corresponds to multiplication by the Riemann zeta function.

GeneralizationsEdit

A related inversion formula more useful in combinatorics is as follows: suppose Template:Math and Template:Math are complex-valued functions defined on the interval Template:Closed-open such that

<math>G(x) = \sum_{1 \le n \le x}F\left(\frac{x}{n}\right)\quad\mbox{ for all }x\ge 1</math>

then

<math>F(x) = \sum_{1 \le n \le x}\mu(n)G\left(\frac{x}{n}\right)\quad\mbox{ for all }x\ge 1.</math>

Here the sums extend over all positive integers Template:Mvar which are less than or equal to Template:Mvar.

This in turn is a special case of a more general form. If Template:Math is an arithmetic function possessing a Dirichlet inverse Template:Math, then if one defines

<math>G(x) = \sum_{1 \le n \le x}\alpha (n) F\left(\frac{x}{n}\right)\quad\mbox{ for all }x\ge 1</math>

then

<math>F(x) = \sum_{1 \le n \le x}\alpha^{-1}(n)G\left(\frac{x}{n}\right)\quad\mbox{ for all }x\ge 1.</math>

The previous formula arises in the special case of the constant function Template:Math, whose Dirichlet inverse is Template:Math.

A particular application of the first of these extensions arises if we have (complex-valued) functions Template:Math and Template:Math defined on the positive integers, with

<math>g(n) = \sum_{1 \le m \le n}f\left(\left\lfloor \frac{n}{m}\right\rfloor\right)\quad\mbox{ for all } n\ge 1.</math>

By defining Template:Math and Template:Math, we deduce that

<math>f(n) = \sum_{1 \le m \le n}\mu(m)g\left(\left\lfloor \frac{n}{m}\right\rfloor\right)\quad\mbox{ for all } n\ge 1.</math>

A simple example of the use of this formula is counting the number of reduced fractions Template:Math, where Template:Mvar and Template:Mvar are coprime and Template:Math. If we let Template:Math be this number, then Template:Math is the total number of fractions Template:Math with Template:Math, where Template:Mvar and Template:Mvar are not necessarily coprime. (This is because every fraction Template:Math with Template:Math and Template:Math can be reduced to the fraction Template:Math with Template:Math, and vice versa.) Here it is straightforward to determine Template:Math, but Template:Math is harder to compute.

Another inversion formula is (where we assume that the series involved are absolutely convergent):

<math>g(x) = \sum_{m=1}^\infty \frac{f(mx)}{m^s}\quad\mbox{ for all } x\ge 1\quad\Longleftrightarrow\quad

f(x) = \sum_{m=1}^\infty \mu(m)\frac{g(mx)}{m^s}\quad\mbox{ for all } x\ge 1.</math>

As above, this generalises to the case where Template:Math is an arithmetic function possessing a Dirichlet inverse Template:Math:

<math>g(x) = \sum_{m=1}^\infty \alpha(m)\frac{f(mx)}{m^s}\quad\mbox{ for all } x\ge 1\quad\Longleftrightarrow\quad

f(x) = \sum_{m=1}^\infty \alpha^{-1}(m)\frac{g(mx)}{m^s}\quad\mbox{ for all } x\ge 1.</math>

For example, there is a well known proof relating the Riemann zeta function to the prime zeta function that uses the series-based form of Möbius inversion in the previous equation when <math>s = 1</math>. Namely, by the Euler product representation of <math>\zeta(s)</math> for <math>\Re(s) > 1</math>

<math>\log\zeta(s) = -\sum_{p\mathrm{\ prime}} \log\left(1-\frac{1}{p^s}\right) = \sum_{k \geq 1} \frac{P(ks)}{k} \iff P(s) = \sum_{k \geq 1} \frac{\mu(k)}{k} \log\zeta(ks), \Re(s) > 1.</math>

These identities for alternate forms of Möbius inversion are found in.<ref>NIST Handbook of Mathematical Functions, Section 27.5.</ref> A more general theory of Möbius inversion formulas partially cited in the next section on incidence algebras is constructed by Rota in.<ref>[On the foundations of combinatorial theory, I. Theory of Möbius Functions|https://link.springer.com/content/pdf/10.1007/BF00531932.pdf]</ref>

Multiplicative notationEdit

As Möbius inversion applies to any abelian group, it makes no difference whether the group operation is written as addition or as multiplication. This gives rise to the following notational variant of the inversion formula:

<math>\mbox{if } F(n) = \prod_{d|n} f(d),\mbox{ then } f(n) = \prod_{d|n} F\left(\frac{n}{d}\right)^{\mu(d)}.</math>

Proofs of generalizationsEdit

The first generalization can be proved as follows. We use Iverson's convention that [condition] is the indicator function of the condition, being 1 if the condition is true and 0 if false. We use the result that

<math>\sum_{d|n}\mu(d)=\varepsilon (n),</math>

that is, <math> 1 * \mu = \varepsilon</math>, where <math>\varepsilon</math> is the unit function.

We have the following:

<math>\begin{align}

\sum_{1\le n\le x}\mu(n)g\left(\frac{x}{n}\right)

&= \sum_{1\le n\le x} \mu(n) \sum_{1\le m\le \frac{x}{n}} f\left(\frac{x}{mn}\right)\\
&= \sum_{1\le n\le x} \mu(n) \sum_{1\le m\le \frac{x}{n}} \sum_{1\le r\le x} [r=mn] f\left(\frac{x}{r}\right)\\
&= \sum_{1\le r\le x} f\left(\frac{x}{r}\right) \sum_{1\le n\le x} \mu(n) \sum_{1\le m\le \frac{x}{n}} \left[m=\frac{r}{n}\right] \qquad\text{rearranging the summation order}\\
&= \sum_{1\le r\le x} f\left(\frac{x}{r}\right) \sum_{n|r} \mu(n) \\
&= \sum_{1\le r\le x} f\left(\frac{x}{r}\right) \varepsilon (r) \\
&= f(x) \qquad\text{since } \varepsilon (r)=0\text{ except when }r=1

\end{align}</math>

The proof in the more general case where Template:Math replaces 1 is essentially identical, as is the second generalisation.

On posetsEdit

Template:See also For a poset Template:Mvar, a set endowed with a partial order relation <math>\leq</math>, define the Möbius function <math>\mu</math> of Template:Mvar recursively by

<math>\mu(s,s) = 1 \text{ for } s \in P, \qquad \mu(s,u) = - \sum_{s \leq t < u} \mu(s,t), \quad \text{ for } s < u \text{ in } P.</math>

(Here one assumes the summations are finite.) Then for <math>f,g: P \to K</math>, where Template:Mvar is a commutative ring, we have

<math>g(t) = \sum_{s \leq t} f(s) \qquad \text{ for all } t \in P</math>

if and only if

<math>f(t) = \sum_{s \leq t} g(s)\mu(s,t) \qquad \text{ for all }t \in P.</math>

(See Stanley's Enumerative Combinatorics, Vol 1, Section 3.7.) The classical arithmetic Mobius function is the special case of the poset P of positive integers ordered by divisibility: that is, for positive integers s, t, we define the partial order <math>s \preccurlyeq t </math> to mean that s is a divisor of t.