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{{short description|Relation between pairs of arithmetic functions}}
{{redirect-distinguish|Möbius transform|Möbius transformation}}
In [[mathematics]], the classic '''Möbius inversion formula''' is a relation between pairs of [[arithmetic function]]s, each defined from the other by sums over [[divisor]]s. It was introduced into [[number theory]] in 1832 by [[August Ferdinand Möbius]].<ref>{{Harvnb|Möbius|1832|pp=105-123}}</ref>

A large generalization of this formula applies to summation over an arbitrary [[Locally finite poset|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 formula==
The classic version states that if {{mvar|g}} and {{mvar|f}} are [[arithmetic function]]s 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 {{mvar|μ}} is the [[Möbius function]] and the sums extend over all positive [[divisor]]s {{mvar|d}} of {{mvar|n}} (indicated by <math>d \mid n</math> in the above formulae). In effect, the original {{math|''f''(''n'')}} can be determined given {{math|''g''(''n'')}} by using the inversion formula. The two sequences are said to be '''Möbius transforms''' of each other.

The formula is also correct if {{mvar|f}} and {{mvar|g}} are functions from the positive integers into some [[abelian group]] (viewed as a {{math|'''Z'''}}-[[module (mathematics)|module]]).

In the language of [[Dirichlet convolution]]s, the first formula may be written as

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

where {{math|∗}} denotes the Dirichlet convolution, and {{math|''1''}} is the [[constant function]] {{math|1=''1''(''n'') = 1}}. The second formula is then written as

:<math>f=\mu * g.</math>

Many specific examples are given in the article on [[multiplicative function]]s.

The theorem follows because {{math|∗}} is (commutative and) associative, and {{math|1=''1'' ∗ ''μ'' = ''ε''}}, where {{mvar|ε}} is the identity function for the Dirichlet convolution, taking values {{math|1=''ε''(1) = 1}}, {{math|1=''ε''(''n'') = 0}} for all {{math|''n'' > 1}}. 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 relations==
Let

:<math>a_n=\sum_{d\mid n} b_d</math>

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 {{math|''ζ''(''s'')}} is the [[Riemann zeta function]].

==Repeated transformations==
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]] {{mvar|φ}}, and repeatedly applies the transformation process, one obtains:

#{{mvar|φ}} the totient function
#{{math|1=''φ'' ∗ ''1'' = ''I''}}, where {{math|1=''I''(''n'') = ''n''}} is the [[identity function]]
#{{math|1=''I'' ∗ ''1'' = ''σ''<sub>1</sub> = ''σ''}}, the [[divisor function]]

If the starting function is the Möbius function itself, the list of functions is:
#{{mvar|μ}}, the Möbius function
#{{math|1=''μ'' ∗ ''1'' = ''ε''}} 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]]
#{{math|1=''ε'' ∗ ''1'' = ''1''}}, the [[constant function]]
#{{math|1=''1'' ∗ ''1'' = ''σ''<sub>0</sub> = d = ''τ''}}, where {{math|1=d = ''τ''}} is the number of divisors of {{mvar|n}}, (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 {{mvar|φ}} 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]].

==Generalizations==
A related inversion formula more useful in [[combinatorics]] is as follows: suppose {{math|''F''(''x'')}} and {{math|''G''(''x'')}} are [[complex number|complex]]-valued [[function (mathematics)|function]]s defined on the [[interval (mathematics)|interval]] {{closed-open|1, ∞}} 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 {{mvar|n}} which are less than or equal to {{mvar|x}}.

This in turn is a special case of a more general form. If {{math|''α''(''n'')}} is an [[arithmetic function]] possessing a [[Dirichlet inverse]] {{math|''α''<sup>−1</sup>(''n'')}}, 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 {{math|1=''α''(''n'') = 1}}, whose [[Dirichlet inverse]] is {{math|1=''α''<sup>−1</sup>(''n'') = ''μ''(''n'')}}.

A particular application of the first of these extensions arises if we have (complex-valued) functions {{math|''f''(''n'')}} and {{math|''g''(''n'')}} 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 {{math|1=''F''(''x'') = ''f''(⌊''x''⌋)}} and {{math|1=''G''(''x'') = ''g''(⌊''x''⌋)}}, 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 fraction]]s {{math|0 < {{sfrac|''a''|''b''}} < 1}}, where {{mvar|a}} and {{mvar|b}} are coprime and {{math|''b'' ≤ ''n''}}. If we let {{math|''f''(''n'')}} be this number, then {{math|''g''(''n'')}} is the total number of fractions {{math|0 < {{sfrac|''a''|''b''}} < 1}} with {{math|''b'' ≤ ''n''}}, where {{mvar|a}} and {{mvar|b}} are not necessarily coprime.  (This is because every fraction {{math|{{sfrac|''a''|''b''}}}} with {{math|1=gcd(''a'',''b'') = ''d''}} and {{math|''b'' ≤ ''n''}} can be reduced to the fraction {{math|{{sfrac|''a''/''d''|''b''/''d''}}}} with {{math|{{sfrac|''b''|''d''}} ≤ {{sfrac|''n''|''d''}}}}, and vice versa.)  Here it is straightforward to determine {{math|1=''g''(''n'') = {{sfrac|''n''(''n'' − 1)|2}}}}, but {{math|''f''(''n'')}} 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 {{math|''α''(''n'')}} is an arithmetic function possessing a Dirichlet inverse {{math|''α''<sup>−1</sup>(''n'')}}:

:<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 notation==
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 generalizations==

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 {{math|''α''(''n'')}} replaces 1 is essentially identical, as is the second generalisation.

==On posets==
{{See also|Incidence algebra}}
For a [[Partially ordered set|poset]] {{mvar|P}}, a set endowed with a partial order relation <math>\leq</math>, define the Möbius function <math>\mu</math> of {{mvar|P}} 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 {{mvar|K}} 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 [[Divisor|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''. 

==Contributions of Weisner, Hall, and Rota==
{{Quotation|
The statement of the general Möbius inversion formula [for partially ordered sets] was first given independently by [[Louis Weisner|Weisner]] (1935) and [[Philip Hall]] (1936); both authors were motivated by group theory problems. Neither author seems to have been aware of the combinatorial implications of his work and neither developed the theory of Möbius functions. In a fundamental paper on Möbius functions, [[Gian-Carlo Rota|Rota]] showed the importance of this theory in combinatorial mathematics and gave a deep treatment of it. He noted the relation between such topics as inclusion-exclusion, classical number theoretic Möbius inversion, coloring problems and flows in networks. Since then, under the strong influence of Rota, the theory of Möbius inversion and related topics has become an active area of combinatorics.<ref>{{Harvnb|Bender|Goldman|1975|pp=789–803}}</ref>
 |sign=|source=}}

==See also==
*[[Farey sequence]]
*[[Inclusion–exclusion principle]]

==Notes==
{{reflist|2}}

==References==
* {{Apostol IANT}}
* {{citation|last1=Bender|first1=Edward A.|last2=Goldman|first2=J.&nbsp;R.|title=On the applications of Möbius inversion in combinatorial analysis|journal=Amer. Math. Monthly|volume=82|year=1975|issue=8|pages=789–803|url=http://www.maa.org/programs/maa-awards/writing-awards/on-the-applications-of-m-bius-inversion-in-combinatorial-analysis|doi=10.2307/2319793|jstor=2319793}}
* {{citation|first1=K.|last1=Ireland|first2=M.|last2=Rosen|title=A Classical Introduction to Modern Number Theory|date=2010|series=Graduate Texts in Mathematics (Book 84)|edition=2nd|publisher=Springer-Verlag|isbn=978-1-4419-3094-1}}
* {{SpringerEOM|id=M/m130180 |title=Möbius inversion |first=Joseph P.S. |last=Kung}}
*{{Citation |last=Möbius |first=A. F. |author-link=August Ferdinand Möbius |year=1832 |title=Über eine besondere Art von Umkehrung der Reihen. |journal=[[Crelle's Journal|Journal für die reine und angewandte Mathematik]] |volume=9 |pages=105–123 |url=https://www.digizeitschriften.de/en/dms/img/?PID=GDZPPN002138654 }}
*{{Citation |last=Stanley |first=Richard P.|year=1997 |url=http://www-math.mit.edu/~rstan/ec/ |title=Enumerative Combinatorics |volume=1  |publisher=Cambridge University Press |isbn=0-521-55309-1}}
*{{Citation |last=Stanley |first=Richard P.|year=1999 |url=http://www-math.mit.edu/~rstan/ec/ |title=Enumerative Combinatorics |volume=2 |publisher=Cambridge University Press |isbn=0-521-56069-1}}

==External links==
*{{MathWorld|MoebiusTransform|Möbius Transform}}

{{DEFAULTSORT:Mobius Inversion Formula}}
[[Category:Arithmetic functions]]
[[Category:Enumerative combinatorics]]
[[Category:Order theory]]

[[ru:Функция Мёбиуса#Обращение Мёбиуса]]