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| In [[mathematics]], the classic '''Möbius inversion formula''' was introduced into [[number theory]] during the 19th century by [[August Ferdinand Möbius]].
| | WHAT can it be? Social media is a phrase used to describe online media websites a particular example is Tube, Facebook, LinkedIn, Twitter, Flickr, Foursquare and ten's of thousands of other ? nternet sites. They are called social media sites because anyone can contribute, comment and share the content placed on these website pages.<br><br>Commercial speech works both verbally and orally. So, yes, it'll be effective via written correspondence. But, because people do business with people they know, like and trust and also the trust factor increases as they get find out you, ultimately you in order to be engage from a live conversation to automatic systems the job.<br><br><br><br>You may be wondering if they should use all the sites available or just a few. And how do you know what go for? The best thing to do is look at the sites and who your target audience is. Where are that they? What type of tools are they using to connect and that's where you'll want to be too.<br><br>All those users that discover these ideas have one goal when social networking: build up their follower list. Many of us like you or your product hence you've came out on top!<br><br>Aggressive push. You'll surely have the means to help your chances of getting a sale if could be more visible in the internet arena. Can be done this your clients' needs your high ticket coaching programs using all effective products promotional tools like Google AdWords, search engine marketing, forum posting, blogging, banner ads, Social Media marketing, etc.<br><br>Don't get me wrong, there are thousands of folks that making n excellent livings working in a home based internet concern. But there are millions struggling to make any money at what. How do you set yourself apart and stay one within the thousands making a good coping?<br><br>You need creativity and imagination if you do intend to succeed and succeed in this buy and sell. You want regarding able present unique yet tasty resources. You can experiment and been released with different decorations and themes. Will need also for you to create diverse of ways. Later on, when you are confident in your skills, you provide to sell in cheap.<br><br>Solution: If you are getting excited, happy and even mad over your track. You are another fool in the town as I found myself. Only allow because they came from are genuine persons as well as not spammy post. You should be smart enough to guage people by their friends' circle, links they post, engagement their own people and more than all reactions.<br><br>If you have any issues about wherever and how to use [http://www.mavsocial.com/ Social Media Management Software], you can contact us at our own web-site. |
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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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WHAT can it be? Social media is a phrase used to describe online media websites a particular example is Tube, Facebook, LinkedIn, Twitter, Flickr, Foursquare and ten's of thousands of other ? nternet sites. They are called social media sites because anyone can contribute, comment and share the content placed on these website pages.
Commercial speech works both verbally and orally. So, yes, it'll be effective via written correspondence. But, because people do business with people they know, like and trust and also the trust factor increases as they get find out you, ultimately you in order to be engage from a live conversation to automatic systems the job.
You may be wondering if they should use all the sites available or just a few. And how do you know what go for? The best thing to do is look at the sites and who your target audience is. Where are that they? What type of tools are they using to connect and that's where you'll want to be too.
All those users that discover these ideas have one goal when social networking: build up their follower list. Many of us like you or your product hence you've came out on top!
Aggressive push. You'll surely have the means to help your chances of getting a sale if could be more visible in the internet arena. Can be done this your clients' needs your high ticket coaching programs using all effective products promotional tools like Google AdWords, search engine marketing, forum posting, blogging, banner ads, Social Media marketing, etc.
Don't get me wrong, there are thousands of folks that making n excellent livings working in a home based internet concern. But there are millions struggling to make any money at what. How do you set yourself apart and stay one within the thousands making a good coping?
You need creativity and imagination if you do intend to succeed and succeed in this buy and sell. You want regarding able present unique yet tasty resources. You can experiment and been released with different decorations and themes. Will need also for you to create diverse of ways. Later on, when you are confident in your skills, you provide to sell in cheap.
Solution: If you are getting excited, happy and even mad over your track. You are another fool in the town as I found myself. Only allow because they came from are genuine persons as well as not spammy post. You should be smart enough to guage people by their friends' circle, links they post, engagement their own people and more than all reactions.
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