Euler Totient Function Sum (Dirchlet Convolution Form)

Using Dirchlet Convolution, the totient sum formula

 

can be re-written more concisely as

in which ` \phi `, ` 1 `  and  ` I `  are arithmetic functions.  This theorem was proven here.

You see, mathematicians like to disguise complicated formulas with simpler-looking ones and they do that by inventing more arcane short-cuts.  Once the formulas have been simplified, they get bored and push the game further by deriving deeper results from these.  So the game seems to be: explore/gather ` \rightarrow ` compress/simplify ` \rightarrow ` explore/gather ` \rightarrow ` compress/simplify, ... and so on and on and on.