TITLE: The sum over zeros in the explicit formula for $\zeta(s)$
QUESTION [8 upvotes]: 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?

REPLY [3 votes]: Edit: After @Lucia's correction/comment, I think the point is that a general qualitative argument based merely on an asymptotic count of zeros cannot directly prove the (conditional) convergence, since such an argument would have to apply too generally, and Lucia's example shows that (presumably some other entire function) could have zeros that conspired to behave badly in the analogue of an "explicit formula".
Failed argument: The argument principle and an application of Jensen's lemma from complex variables, and Stirling's asymptotic for Gamma, via the functional equation for zeta, show that the number of zeros up to height $T$ is bounded by a constant times $T\log T$ (in fact giving asymptotics...). This, together with the symmetry of zeros from the functional equation and so on, gives the conditional convergence.
That is, the additional details that get used in a proof of an explicit formula are probably necessary.