So I have this equality to prove for any $n \in \mathbb{N}$:
$$ \sum_{d|n} \sigma(d) \phi(\frac{n}{d}) = n \tau(n) $$
So I was able to show that left and right side are multiplicative.
So how can I prove that the left side is equal to the right.
So if it holds for $n=1$, $n=2$
Then I take:
$n=p_1^{\alpha_1}...p_k^{\alpha_k}$ And have to see if it works for this.
But am not sure how do I represent the left side, since I don't know how this actually affects the sum $\phi(\frac{n}{d})$.
Any help, answers or hints would be appreciated.
Thank you in advance.