I found on Wikipedia that $$D(s,\sigma_a \sigma_b) := \sum_{n \geqslant 1} \frac{\sigma_a(n)\sigma_b(n)}{n^s} = \frac{\zeta(s)\zeta(s-a)\zeta(s-b)\zeta(s-a-b)}{\zeta(2s-a-b)}$$
where as usual $\sigma_k(n) = \sum_{d | n} d^k$ is the $k$-th power divisors function. I do understand how to reach the Dirichlet series for $\sigma_k$, however I do not understand why is that so for the product.