Can it be said that the following equivalence is true?
$$ 0 = \sum_j^\infty (e^{\zeta_j} - e^{-\zeta_j}) \Leftrightarrow \sum_j^\infty e^{\zeta_j} = \sum_j^\infty e^{-\zeta_j} \quad , \quad \zeta_j \in \mathbb{C}$$
Firstly, for the finite case, this is obviously true, but I'm not convinced that this necessarily carries over into the infinite case, as the left-hand side is not convergent (proportional to complex hyperbolic sine, which isn't convergent).
Formally, the relation seems true, at least in the right to left inplication as the number of summations should be of equal number. Left to right, however, I'm not sure how to justify the left to right implication though. Cauchy's convergence criterion doesn't apply because complex sinh is not even conditionally convergent.
The $\zeta_j$s are rapidities of particle states and $j$ is merely an index number. The idea being to be able to find the energy density via a seiries expansion and integral. They (theoretically) range over the entire complex plane and there is no relation between $\zeta_i$ and $\zeta_j$, where $i \neq j$.