Editing Möbius inversion formula (section)

==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>