I wanted to prove that $\varphi$ is multiplicative (that for $(a,b)=1$, $\varphi(ab)=\varphi(a)\varphi(b)$) using the following idea:
- Define $\varphi'$ by $n = \varphi'(n) + \varphi(n)$.
- Multiply out $\varphi(a)\varphi(b) = (a - \varphi'(a))(b - \varphi'(b)) = ab - \varphi'(a)b - a \varphi'(b) + \varphi'(a)\varphi'(b)$
- Use the principle of inclusion exclusion ($|A \cup B| = |A| + |B| - |A \cap B|$).
but I could not get it to work out.
Is this approach possible at all? How can it be saved?
So I just need to construct sets $A(a,b)$ and $B(a,b)$ that satisfy the following
$$\begin{matrix} \varphi'(ab) &=& \varphi'(a) b &+& a \varphi'(b) &-& \varphi'(a)\varphi'(b)\\ || & & || & & || & & || \\ |A \cup B| &=& |A| &+& |B| &-& |A \cap B| \end{matrix}$$
then the proof follows from simple algebra. I just cannot find any way to construct such sets - since the identity is true I imagine they do exist but I am not be completely sure especially since I couldn't construct them.