I am trying to solve the following problem.
Prove that: $$\sum_{t\mid n} d(t)^3= \left(\sum_{t\mid n}d(t)\right)^2$$
I know that there exists the following identity: $$\sum i^3=\left (\sum i\right)^2$$
What is this identity called? Does anyone have an approach to this problem?
Why can't I just substitute $d(t)$ for $i$?
I have thought about using $\sigma_\alpha(p^\alpha)$ but I am unsure of the correct way to proceed. This is problem $2.12$ from Apostol's Analytic Number Theory.