Theorem 3.8 (Partial Fraction approximation of $z e t a^{'} ∕ z e t a$ ).

(i) Let $s = σ + i t$ with $| σ | \leq 10$ , $s \neq 1$ and $ζ (s) \neq 0$ . Then
$\frac{ζ^{'} (s)}{ζ (s)} = - \frac{1}{s - 1} + \sum_{| ρ - s | \leq \frac{1}{10}} \frac{1}{s - ρ} + O (\log (| t | + 2)) .$

where the sum is over the zeroes $ρ$ of $ζ$ counted with multiplicity.
(ii) For any $T \geq 0$ , there are $≪ \log (T + 2)$ many zeroes $ρ$ of $ζ$ (counted with multiplicity) with $| Im (ρ) | \in [T, T + 1]$ .