Transforming a sum of products in a product of sums

Question

I am studying the proof for the formula of $\sigma (n)$, the divisor function. At a certain point there is this equivalence but I can't see how it works:

$$\sum_{b_i \in \{0,1,\dotsc a_i\}}{p_1^{b_1} \cdot p_2^{b_2} \dotsm p_k^{b_k}} = \prod_{i=1}^{k}{(1+p_i+p_i^{2}+ \dotsc+p_i^{a_i})}.$$

How is it possible to transform a sum like this in a product of sums? Perhaps there is some combinatorics behind this but I don't see it.

Thanks for your help.

Answer 1

The sum in the product, is the sum of divisors of $p_i$ the product of these sums contain every product without repetition, of their divisors not exceeding the original number.