Suppose that $z$ is some complex value. Is it possible to prove that

$$\lim_{n \rightarrow \infty} \sum_{j = 1}^n {\sqrt{n \over j}} \cdot \cosh(z \log {n \over j}-\operatorname{ Arccoth} (2z)) $$

cannot converge if $\Re(z) <> 0$?

Empirical testing strongly suggests both that it doesn't converge, and that the bounds of both the real and imaginary parts of the limit grow without bound as n increases, but I'd be really curious if there's a way to prove this.