Suppose we have the zeta function $\zeta(s)$, and we want to multiply it by its complex conjugate $\zeta(s)^*$.
Since $\zeta(s)^* = \zeta(s^*)$, we get
$\displaystyle \zeta(s)\cdot\zeta(s)^* = \left[ \sum_p \frac{1}{p^s} \right] · \left[ \sum_q \frac{1}{q^{s^*}} \right]$
I'm trying to see if it's justified to transform the above into the following:
$\displaystyle \zeta(s)\cdot\zeta(s)^* = \sum_{p,q} \frac{1}{p^s \cdot q^{s^*}}$
where the summation is over all $p, q \in \Bbb N$.
But will that fly, at least for some region of s (like $Re\{s\} > 1$)?
I don't understand how to multiply Dirichlet series together where the argument is different ($s$ vs $s^*$ in this case). However, I don't need to prove it for all Dirichlet series, just zeta. Also, if they're unequal, I'd love to know whether one bounds the other!
In any region where you have absolute convergence of each of the two series, you can write it in that form. In short, this is because you are able to rearrange terms in absolutely convergent series.
You also ask about using a region like when $\Re s > 1$. Yes - that'll do. But it also absolutely wouldn't make sense to talk about it otherwise, as neither series $\displaystyle \sum_n \frac{1}{n^s}$ converges at all when $\Re s \leq 1$. While we might talk about their analytic continuations, this is different then talking about the series itself.