Local zero count for the Riemann zeta function ($N(t+1)-N(t)\ll \log (|t|+3)$)

The number of Nontrivial zeros of the Riemann zeta function in a height interval of length one is $O(\log (|t|+3))$, including multiplicity and either endpoint. Subtract the Riemann–von Mangoldt formula at the endpoints and use conjugation at negative heights. Summing these counts with reciprocal-height weights gives ${\sum }_{|\Im \rho |\le T}1/|\rho |\ll {\log }^2(T+3)$.