Editing Dirichlet convolution (section)

===Other formulas===

{| class="wikitable" border="1"
! Arithmetic function !! Dirichlet inverse:<ref>See Apostol Chapter 2.</ref>
|-
| Constant function with value 1 ||[[Möbius function]] ''μ''
|-
| <math>n^{\alpha}</math> || <math>\mu(n) \,n^\alpha</math>
|-
| [[Liouville's function]] ''λ'' || Absolute value of Möbius function {{abs|''μ''}}
|-
| [[Euler's totient function]] <math>\varphi</math> ||<math>\sum_{d|n} d\, \mu(d)</math>
|-
| The [[sum of divisors|generalized sum-of-divisors function]] <math>\sigma_{\alpha}</math> || <math>\sum_{d|n} d^{\alpha} \mu(d) \mu\left(\frac{n}{d}\right)</math> 
|}

An exact, non-recursive formula for the Dirichlet inverse of any [[arithmetic function]] ''f'' is given in [[Divisor sum identities#The Dirichlet inverse of an arithmetic function|Divisor sum identities]]. A more [[partition theory|partition theoretic]] expression for the Dirichlet inverse of ''f'' is given by 

:<math>f^{-1}(n) = \sum_{k=1}^{\Omega(n)} \left\{ 
     \sum_{{\lambda_1+2\lambda_2+\cdots+k\lambda_k=n} \atop {\lambda_1, \lambda_2, \ldots, \lambda_k | n}} 
     \frac{(\lambda_1+\lambda_2+\cdots+\lambda_k)!}{1! 2! \cdots k!} 
     (-1)^k f(\lambda_1) f(\lambda_2)^2 \cdots f(\lambda_k)^k\right\}.</math>
The following formula provides a compact way of expressing the Dirichlet inverse of an invertible arithmetic function ''f'' :

<math>f^{-1}=\sum_{k=0}^{+\infty}\frac{(f(1)\varepsilon-f)^{*k}}{f(1)^{k+1}}</math>

where the expression <math>(f(1)\varepsilon-f)^{*k}</math> stands for the arithmetic function <math>f(1)\varepsilon-f</math> convoluted with itself ''k'' times. Notice that, for a fixed positive integer <math>n</math>, if <math>k>\Omega(n)</math> then <math>(f(1)\varepsilon-f)^{*k}(n)=0</math> , this is because <math>f(1)\varepsilon(1) - f(1) = 0</math> and every way of expressing ''n'' as a product of ''k'' positive integers must include a 1, so the series on the right hand side converges for every fixed positive integer ''n.''