The Dirichlet Convolution has the following property:-
The proof is a straightforward exercise.
Proof:-
For any ` n ` we have
` [f \mbox{*} (g + h)](n) `
= ` \sum_{d|n}f(d) \cdot (g+h)(n/d) `
= ` \sum_{d|n}\{ f(d) \cdot g(n/d) + f(d) \cdot h(n/d) \}`
= ` \sum_{d|n}f(d) \cdot g(n/d) + \sum_{d|n}f(d) \cdot h(n/d) `
= ` (f \mbox{*}g)(n) + (f \mbox{*}h)(n) `
= ` (f \mbox{*}g + f \mbox{*}h)(n) `
(proven)
