I was wondering if explicit closed forms for rearrangements of $\eta(s)$, for $s$ such that the series is not absolutely convergent, are useful in studying the Dirichlet $\eta$ function.
I am asking this specific question because although we learn about rearrangements of conditionally convergent series, I haven't seen any good applications of such facts, in particular for rearrangements we can compute explicitly.
By Riemann's theorem any conditionally convergent series can be rearranged to converge to any given value (or diverge). So, in general, there is not much you could do. However, if you impose a particular order of summation (i.e. the principal value), and define $\eta(s)$ with this order of summation, then, in principle, you could derive some useful facts. For example, the Eisenstein series is not absolutely convergent for $r = 2$, but if we impose a particular order of summation, we can obtain its Fourier expansion which will be absolutely convergent for $r = 2$.