Editing Dirichlet convolution (section)

==Dirichlet inverse==

===Examples===

Given an arithmetic function <math>f</math> its Dirichlet inverse <math>g = f^{-1} </math> may be calculated recursively: the value of <math>g(n) </math> is in terms of <math>g(m)</math> for <math>m<n</math>.

For <math>n=1</math>:
:<math>(f * g) (1) = f(1) g(1) = \varepsilon(1) = 1 </math>,  so
:<math>g(1) = 1/f(1)</math>. This implies that <math>f</math> does not have a Dirichlet inverse if <math>f(1) = 0</math>.

For <math>n=2</math>:
:<math>(f * g) (2) = f(1) g(2) + f(2) g(1) = \varepsilon(2) = 0</math>,
:<math>g(2) = -(f(2) g(1))/f(1) </math>,

For <math>n=3</math>:
: <math>(f * g) (3) = f(1) g(3) + f(3) g(1) = \varepsilon(3) = 0</math>,
: <math>g(3) = -(f(3) g(1))/f(1) </math>,

For <math>n=4</math>:
: <math>(f * g) (4) = f(1) g(4) + f(2) g(2) + f(4) g(1) = \varepsilon(4) = 0</math>,
: <math>g(4) = -(f(4) g(1) + f(2) g(2))/f(1)  </math>,

and in general for <math>n>1</math>,

:<math>
g(n) \ =\ 
\frac {-1}{f(1)} \mathop{\sum_{d\,\mid \,n}}_{d < n}
f\left(\frac{n}{d}\right) g(d).
</math>

===Properties===

The following properties of the Dirichlet inverse hold:<ref>Again see Apostol Chapter 2 and the exercises at the end of the chapter.</ref>

* The function ''f'' has a Dirichlet inverse if and only if {{nowrap|''f''(1) ≠ 0}}. 
* The Dirichlet inverse of a [[multiplicative function]] is again multiplicative. 
* The Dirichlet inverse of a Dirichlet convolution is the convolution of the inverses of each function: <math>(f \ast g)^{-1} = f^{-1} \ast g^{-1}</math>. 
* A multiplicative function ''f'' is [[completely multiplicative]] if and only if <math>f^{-1}(n) = \mu(n) f(n)</math>.
* If ''f'' is [[completely multiplicative]] then <math>(f \cdot g)^{-1} = f \cdot g^{-1}</math> whenever <math>g(1) \neq 0</math> and where <math>\cdot</math> denotes pointwise multiplication of functions.

===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.''