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 ${\sum }_{|\Im \rho |\le T}1/|\rho |\ll {\log }^2(T+3)$.

Put $F(s)=(s-1)\zeta (s)$. The fractional-part continuation formula for the Riemann zeta function gives $|F(s)|\ll (|s|+1{)}^2$ for $\mathrm{Re}s\ge 1/4$. At $2+ij$, the reciprocal Euler product bounds $|\zeta (2+ij)|$ below by $1/\zeta (2)$. Applying the Jensen zero-count bound with outer radius $7/4$ and inner radius $8/5$ gives $O(\log (|j|+2))$ zeros in the latter disk. Those disks cover the half-strip $1/2\le \mathrm{Re}s\le 1$. Reflecting zeros using the functional equation of the Riemann zeta function completes the $O(T\log T)$ bound without requiring left-half-plane growth estimates.

Evaluate the real logarithmic derivative of Riemann xi at real part two, where every summand is positive and comparable to this kernel. The gamma logarithmic derivative and the bounded zeta logarithmic derivative give the logarithmic bound. It implies the local unit-interval count, and conversely such local counts imply this smoothed bound by summing the decaying tails.