Im sure this question is completely trivial, but I just want to check my understanding:
For the various explicit formulae in Analytic Number Theory involving sums over the nontrivial zeros of $\zeta$ how are the zeros evaluated?
For example, take $\psi(x)=x-\sum_{\zeta(\rho)=0}\frac{x^\rho}\rho -\log(2\pi)$. Wikipedia states "(The sum is not absolutely convergent, so we take the zeros in order of the absolute value of their imaginary part.)"
Does this mean that we take some limit to infinity, $\lim_{l\to\infty}$, and every time we get to a point where there is a zero with $\operatorname{Im}(l)$ we sum? So, for example, if there is a zero off of the $\operatorname{Re}(1/2)$ line the terms above would be: $$\frac{x^{\rho_1}}{\rho_1}+\frac{x^{\rho_2}}{\rho_2}+\frac{x^{\rho_3}}{\rho_3}+\frac{x^{\rho_4}}{\rho_4},$$ where the four $\rho$s represent the four zeros with the same imaginary part (conjugate symmetry as well as reflection about the $1/2$ line). So there are four independent expressions evaluated at the same time?
Just to make an official answer of the point @reuns makes in a comment, and perhaps also relate it to more elementary definitional things:
As reuns says, the limit in question is $\lim_{T\to+\infty}\sum_{|\rho|<T} {x^\rho\over \rho}$, (with multiplicities) from the way the (von Mangoldt form of the) Explicit Formula is derived. First, note that an unconditional convergence assertion would involve two bounds: $\lim_{T_1,T_2\to+\infty} \sum_{-T_1<\Im(\rho)<T_2}{x^\rho\over \rho}$. We do not need to know whether or not this converges in order to be able to refer to the specific limit in the actual Explicit Formula. (This is vaguely similar to Cauchy Principal Value integrals, which are "improper", and do not at all converge, but with a "summation/integration" convention do converge.)
Yes, if $\rho$ is a $0$, then so are $1-\rho$ and $\overline{\rho}$. This, and Hadamard's theorem about global growth of entire functions versus convergence of sums over zeros, does guaranteed convergence of that summation-convention version of the sum over zeros.
But/and, again, the indicated summation-convention version of the infinite sum is a single-parameter limit (as in Cauchy Principal Value integrals), while the "true/full" infinite/improper sum would use two independent limits.