The explicit formula for $\zeta(s)$ is: $$ \psi(x)=x-\sum_{|\operatorname{Im}\rho|<T}\frac{x^\rho}{\rho}-\log(2\pi)-\log\left(1-\frac{1}{x^2}\right)+O\left(\frac{x\log^2T}{T}\right), $$ where $\psi(x)=\sum_{p^k<x}\log p$, $x>1$ is a non-integer, $\rho$ is a nontrivial zero of $\zeta(s)$, and the sum over $\rho$ is taken with multiplicities.

Letting $T\rightarrow\infty$, we can conclude that the sum over all nontrivial zeros $\rho$ is convergent (albeit conditionally) since the other terms in the formula are finite. Is there a way to *directly* show that this sum is convergent?