Editing Möbius inversion formula (section)

==Repeated transformations==
Given an arithmetic function, one can generate a bi-infinite sequence of other arithmetic functions by repeatedly applying the first summation. 

For example, if one starts with [[Euler's totient function]] {{mvar|φ}}, and repeatedly applies the transformation process, one obtains:

#{{mvar|φ}} the totient function
#{{math|1=''φ'' ∗ ''1'' = ''I''}}, where {{math|1=''I''(''n'') = ''n''}} is the [[identity function]]
#{{math|1=''I'' ∗ ''1'' = ''σ''<sub>1</sub> = ''σ''}}, the [[divisor function]]

If the starting function is the Möbius function itself, the list of functions is:
#{{mvar|μ}}, the Möbius function
#{{math|1=''μ'' ∗ ''1'' = ''ε''}} where <math display="block">\varepsilon(n) = \begin{cases} 1, & \text{if }n=1 \\ 0, & \text{if }n>1 \end{cases} </math> is the [[unit function]]
#{{math|1=''ε'' ∗ ''1'' = ''1''}}, the [[constant function]]
#{{math|1=''1'' ∗ ''1'' = ''σ''<sub>0</sub> = d = ''τ''}}, where {{math|1=d = ''τ''}} is the number of divisors of {{mvar|n}}, (see [[divisor function]]).

Both of these lists of functions extend infinitely in both directions.  The Möbius inversion formula enables these lists to be traversed backwards.

As an example the sequence starting with {{mvar|φ}} is:

:<math>f_n =
  \begin{cases}
   \underbrace{\mu * \ldots * \mu}_{-n \text{ factors}} * \varphi & \text{if } n < 0 \\[8px]
   \varphi & \text{if } n = 0 \\[8px]
   \varphi * \underbrace{\mathit{1}* \ldots * \mathit{1}}_{n \text{ factors}} & \text{if } n > 0
  \end{cases}
</math>

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]].