Dirichlet Convolution Identities

Question

I am unsure of the proof of the identity involving the identity arithmetic function, $e$. That is, the identity $f\ast e=f$. My proof so far is: $$[f\ast e](n)=\sum\limits_{d\mid n}f(d)e(\frac{n}{d})$$ Letting $d=n$ to make $e\neq0$: $$\sum\limits_{d\mid n}f(d)e(\frac{n}{d})=f(n)$$ The step I don't understand is the part where $e$ CAN'T equal 0. I know it's nice when we say is doesn't equal zero, but what lets us do this?

Answer 1

By definition, $e(n)=0$ if $n>1$ and $e(1)=1$.

So, in the sum $\sum\limits_{d\mid n}f(d)e(\frac{n}{d})$, the only non-zero term is when $\frac{n}{d}=1$.