Dirchlet Convolution is Distributive over Addition

The Dirichlet Convolution has the following property:-

The proof is a straightforward exercise.


Proof:-

For any ${\displaystyle n}$ we have
   ${\displaystyle \left[f\text{*}\left(g+h\right)\right]\left(n\right) }$
= ${\displaystyle \sum _{d|n}f\left(d\right)\cdot \left(g+h\right)\left(\frac{n}{d}\right)}$
= ${\displaystyle \sum _{d|n}\{f\left(d\right)\cdot g\left(\frac{n}{d}\right)+f\left(d\right)\cdot h\left(\frac{n}{d}\right)\}}$
= ${\displaystyle \sum _{d|n}f\left(d\right)\cdot g\left(\frac{n}{d}\right)+\sum _{d|n}f\left(d\right)\cdot h\left(\frac{n}{d}\right)}$
= ${\displaystyle \left(f\text{*}g\right)\left(n\right)+\left(f\text{*}h\right)\left(n\right) }$
= ${\displaystyle \left(f\text{*}g+f\text{*}h\right)\left(n\right) }$

(proven)