The following is a screenshot from a paper by Daniel B. Grunberg called On asymptotics, Stirling numbers, Gamma function, and polylogs.
I only offer the page as a reference to explain equation 3.1. Don't worry about the rest of the paper, though interesting.
Now for some sequence $A=\{a_1, a_2, a_3,... a_n\}$ let us define the function $$f_n(s)=\sum_{k=1}^n{\frac{1}{{a_k}^s}}$$
In equation 3.1 replace the harmonic number with $f_n(s\cdot r_j)$ and remove the alternating sign to define a new function:
$$Q_{r,1,n}(s)=\sum_{\{r\}}\prod_{j=1}^l\frac{(-1)^{i_j}}{i_j!}\left(\frac{f_n(s\cdot r_j)}{r_j}\right)^{i_j}$$
$Q_{0,1,n}(s)=1$.
Prove the following identities. $$\sum_{r=0}^n Q_{r,1,n}(s)=\prod_{k=1}^n\left(1-\frac{1}{{a_k}^s}\right)$$ $$Q_{n,1,n}(s)=(-1)^{n}\frac{\sum_{k=1}^n {a_k}^s}{\prod_{k=1}^n {a_k}^s}$$ $$Q_{n-1,1,n}(s)=(-1)^{n-1}\frac{1}{\prod_{k=1}^n {a_k}^s}$$
I discovered these identities, but it's been so long I can't remember how I came to them. Some professional proof-work would be helpful.
Also how would we continue such expressions of $Q_{n-k,1,n}(s)$ for $k=2,3,...,(n-1)$?