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| In [[mathematics]], the classic '''Möbius inversion formula''' was introduced into [[number theory]] during the 19th century by [[August Ferdinand Möbius]].
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| Other Möbius inversion formulas are obtained when different [[Locally finite poset|locally finite partially ordered set]]s replace the classic case of the natural numbers ordered by divisibility; for an account of those, see [[incidence algebra]].
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| ==Statement of the formula==
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| The classic version states that if ''g'' and ''f'' are [[arithmetic function]]s satisfying
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| : <math>g(n)=\sum_{d\,\mid \,n}f(d)\quad\text{for every integer }n\ge 1</math>
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| then
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| :<math>f(n)=\sum_{d\,\mid\, n}\mu(d)g(n/d)\quad\text{for every integer }n\ge 1</math>
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| where μ is the [[Möbius function]] and the sums extend over all positive [[divisor]]s ''d'' of ''n''. In effect, the original ''f''(''n'') can be determined given ''g''(''n'') by using the inversion formula. The two sequences are said to be [[Möbius transform]]s of each other.
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| The formula is also correct if ''f'' and ''g'' are functions from the positive integers into some [[abelian group]] (viewed as a <math>\mathbb{Z}</math>-[[module (mathematics)|module]]).
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| In the language of [[Dirichlet convolution]]s, the first formula may be written as
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| :<math>g=f*1</math>
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| where ''*'' denotes the Dirichlet convolution, and ''1'' is the [[constant function]] <math>1(n)=1</math>. The second formula is then written as | |
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| :<math>f=\mu * g.</math>
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| Many specific examples are given in the article on [[multiplicative function]]s.
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| The theorem follows because ''*'' is (commutative and) associative, and 1''*''μ=''i'', where ''i'' is the identity function for the Dirichlet convolution, taking values ''i''(1)=1, ''i''(''n'')=0 for all ''n''>1. Thus μ''*g'' = μ''*''(1''*f'') = (μ''*''1)''*f'' = ''i*f'' = ''f''.
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| ==Repeated transformations==
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| Given an arithmetic function, one can generate a bi-infinite sequence of other arithmetic functions by repeatedly applying the first summation.
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| For example, if one starts with [[Euler's totient function]] <math>\varphi</math>, and repeatedly applies the transformation process, one obtains:
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| #<math>\varphi</math> the totient function
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| #<math>\varphi*1=\operatorname{Id}</math> where <math>\operatorname{Id}(n)=n</math> is the [[identity function]]
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| #<math>\operatorname{Id} *1 =\sigma_1 =\sigma</math>, the [[divisor function]]
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| If the starting function is the Möbius function itself, the list of functions is:
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| #<math>\mu</math>, the Möbius function
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| #<math>\mu*1 = \varepsilon</math> where <math>\varepsilon(n) = \begin{cases} 1, & \mbox{if }n=1 \\ 0, & \mbox{if }n>1 \end{cases} </math> is the [[unit function]]
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| #<math>\varepsilon*1 = 1 </math>, the [[constant function]]
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| #<math>1*1=\sigma_0=\operatorname{d}=\tau</math>, where <math>\operatorname{d}=\tau</math> is the number of divisors of ''n'', (see [[divisor function]]).
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| Both of these lists of functions extend infinitely in both directions. The Möbius inversion formula enables these lists to be traversed backwards. 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]].
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| ==Generalizations==
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| {{See also|Incidence algebra}}
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| A related inversion formula more useful in [[combinatorics]] is as follows: suppose ''F''(''x'') and ''G''(''x'') are [[complex number|complex]]-valued [[function (mathematics)|function]]s defined on the [[interval (mathematics)|interval]] <nowiki>[1,∞)</nowiki> such that
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| :<math>G(x) = \sum_{1 \le n \le x}F(x/n)\quad\mbox{ for all }x\ge 1</math>
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| then
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| :<math>F(x) = \sum_{1 \le n \le x}\mu(n)G(x/n)\quad\mbox{ for all }x\ge 1.</math>
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| Here the sums extend over all positive integers ''n'' which are less than or equal to ''x''.
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| This in turn is a special case of a more general form. If <math>\alpha(n)</math> is an [[arithmetic function]] possessing a [[Dirichlet inverse]] <math>\alpha^{-1}(n)</math>, then if one defines
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| :<math>G(x) = \sum_{1 \le n \le x}\alpha (n) F(x/n)\quad\mbox{ for all }x\ge 1</math>
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| then
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| :<math>F(x) = \sum_{1 \le n \le x}\alpha^{-1}(n)G(x/n)\quad\mbox{ for all }x\ge 1.</math>
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| The previous formula arises in the special case of the constant function <math>\alpha(n)=1</math>, whose [[Dirichlet inverse]] is <math>\alpha^{-1}(n)=\mu(n)</math>.
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| A particular application of the first of these extensions arises if we have (complex-valued) functions ''f''(''n'') and ''g''(''n'') defined on the positive integers, with
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| :<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>
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| By defining <math>F(x) = f(\lfloor x\rfloor)</math> and <math>G(x) = g(\lfloor x\rfloor)</math>, we deduce that
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| :<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>
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| A simple example of the use of this formula is counting the number of [[reduced fraction]]s 0 < ''a''/''b'' < 1, where ''a'' and ''b'' are coprime and ''b''≤''n''. If we let ''f''(''n'') be this number, then ''g''(''n'') is the total number of fractions 0 < ''a''/''b'' < 1 with ''b''≤''n'', where ''a'' and ''b'' are not necessarily coprime. (This is because every fraction ''a''/''b'' with gcd(''a'',''b'') = ''d'' and ''b''≤''n'' can be reduced to the fraction (''a''/''d'')/(''b''/''d'') with ''b''/''d'' ≤ ''n''/''d'', and vice versa.) Here it is straightforward to determine ''g''(''n'') = ''n''(''n''-1)/2, but ''f''(''n'') is harder to compute.
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| Another inversion formula is (where we assume that the series involved are absolutely convergent):
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| :<math>g(x) = \sum_{m=1}^\infty \frac{f(mx)}{m^s}\quad\mbox{ for all } x\ge 1\quad\Longleftrightarrow\quad
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| f(x) = \sum_{m=1}^\infty \mu(m)\frac{g(mx)}{m^s}\quad\mbox{ for all } x\ge 1.</math>
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| As above, this generalises to the case where <math>\alpha(n)</math> is an arithmetic function possessing a Dirichlet inverse <math>\alpha^{-1}(n)</math>:
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| :<math>g(x) = \sum_{m=1}^\infty \alpha(m)\frac{f(mx)}{m^s}\quad\mbox{ for all } x\ge 1\quad\Longleftrightarrow\quad
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| f(x) = \sum_{m=1}^\infty \alpha^{-1}(m)\frac{g(mx)}{m^s}\quad\mbox{ for all } x\ge 1.</math>
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| ==Multiplicative notation==
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| 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:
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| : <math>
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| \mbox{If } F(n) = \prod_{d|n} f(d),\mbox{ then } f(n) = \prod_{d|n} F(n/d)^{\mu(d)}. \,
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| </math>
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| ==Proofs of generalizations==
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| 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)=i(n)</math>, that is, 1*μ=''i''.
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| We have the following:
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| <math>\begin{align}
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| \sum_{1\le n\le x}\mu(n)g\left(\frac{x}{n}\right)
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| &= \sum_{1\le n\le x} \mu(n) \sum_{1\le m\le x/n} f\left(\frac{x}{mn}\right)\\
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| &= \sum_{1\le n\le x} \mu(n) \sum_{1\le m\le x/n} \sum_{1\le r\le x} [r=mn] f\left(\frac{x}{r}\right)\\
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| &= \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 x/n} [m=r/n] \qquad\text{rearranging the summation order}\\
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| &= \sum_{1\le r\le x} f\left(\frac{x}{r}\right) \sum_{n|r} \mu(n) \\
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| &= \sum_{1\le r\le x} f\left(\frac{x}{r}\right) i(r) \\
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| &= f(x) \qquad\text{since }i(r)=0\text{ except when }r=1
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| \end{align}</math>
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| The proof in the more general case where α(''n'') replaces 1 is essentially identical, as is the second generalisation.
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| ==See also==
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| * [[Lambert series]]
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| * [[Farey sequence]]
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| ==References==
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| * {{Apostol IANT}}
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| * {{SpringerEOM|id=M/m130180 |title=Möbius inversion |first=Joseph P.S. |last=Kung}}
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| *K. Ireland, M. Rosen. ''A Classical Introduction to Modern Number Theory'', (1990) Springer-Verlag.
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| {{reflist}}
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| ==External links==
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| {{ProofWiki|id=Möbius_Inversion_Formula|title=Möbius Inversion Formula}}
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| {{DEFAULTSORT:Mobius Inversion Formula}}
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| [[Category:Arithmetic functions]]
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| [[Category:Enumerative combinatorics]]
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| [[Category:Order theory]]
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| [[ru:Функция Мёбиуса#Обращение Мёбиуса]]
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Hello and welcome. My name is Numbers Wunder. Minnesota is exactly where he's been residing for many years. One of the very very best things in the world for me is to do aerobics and I've been doing it for quite a whilst. Hiring is his profession.
Visit my site; tube.ledawy.com