Local zero count for the Riemann zeta function
= Local zero count for the Riemann zeta function
{title2=$N(t+1)-N(t)\ll\log(|t|+3)$}
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 $\sum_{|\Im\rho|\le T}1/|\rho|\ll\log^2(T+3)$.