**Definition:** Let $n=\prod\_k p\_k^{c\_k}$, with $p\_k \in \mathbb P$ and
$$
A(n)=\sum\_{d|n} \mu(d)\Lambda(d)=\sum \log p\_k^{1-\delta\_{0,c\_k}} ,
$$

> 
> with the [Möbius function](https://en.wikipedia.org/wiki/M%C3%B6bius_function#Definition) $\mu(n)$, which is:
> 
> 
> * $μ(n) = \pm 1$ if $n$ is a square-free positive integer with an even/odd number of prime factors.
> * $μ(n) = 0$ if $n$ has a squared prime factor.
> 
> 
> And the [von Mangoldt function](https://en.wikipedia.org/wiki/Von_Mangoldt_function) $Λ(n)$, defined as
> $$
>  \Lambda(n) = \begin{cases} \log p & \text{if }n=p^k \text{ for some prime } p \text{ and integer } k \ge 1, \\ 0 & \text{otherwise.} \end{cases} 
> $$
> 
> 
> 

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**Properties of $A(n)$:**

* $A(\cdot)$ basically takes every number $n$ to a sum of logarithms of prime factors, *e.g.*
$$
A(12)=A(2^23)=\log 2 + \log 3=A(6);\\
A(4)=A(2)=\log 2.
$$
* The arithmetic function is [additive](https://en.wikipedia.org/wiki/Additive_function), since $A(nm)=A(n)+A(m)$ only for co-prime $n$ and $m$, *e.g.*
$$
\log 2 + \log 3=A(6)=A(2\cdot 3) = A(2)+A(3)= \log 2 + \log 3
$$
* You can apply the [Möbius Inversion](https://en.wikipedia.org/wiki/M%C3%B6bius_inversion_formula#Statement_of_the_formula) to it. You'll get:
$$
\mu(n)\Lambda(n)=\sum\_{d|n} \mu(d) A(n/d),
$$
*e.g.* (where I write only terms with $\mu(d)\neq 0$) $$0=\mu(12)\Lambda(12)=\mu(6)A(2)+\mu(3)A(4)+\mu(2)A(6)+\mu(1)A(12)\\ =\log 2-\log 2-(\log 2+\log 3)+(\log 2+\log 3)=0$$

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> 
> **Was the additive arithmetic function $A(n)$ ever used in any context?**
> 
> 
>