Editing Möbius inversion formula (section)

==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>