This is just a note. 

There is a Riemann sum involved here:
$$\lim_{n\rightarrow\infty}R_n(z):=\lim_{n\rightarrow\infty}\,\,\frac1n\sum_{j=1}^n\sqrt\frac{n}{j}\sin\left(z\log{\frac{n}{j}}\right)
=-\int_0^1\frac{\sin(z\log x)}{\sqrt{x}}\,dx=\frac{2z}{1+4z^2};$$
valid (at least) for $z\in\mathbb{R}$. In view of this, the OP's question amounts to: how fast does the Riemann sums converge to the integral with respect to $n$; that is,
$$R_n(z)-\int_0^1\frac{-\sin(z\log x)}{\sqrt{x}}\,dx=O(??) \tag1$$
Some numerical evidence suggests that (even for real $z\neq0$) the order of convergence is weaker than $\frac1n$, which means the OP's limit does not converge.