Theorem 3.11 (Bounding $z e t a^{'} ∕ z e t a$ ). Assuming that:

$c > 0$ sufficiently small
$T \geq 0$
$Re (s) \geq - 10$
$| Im (s) | \in [T, T + 1]$
$s$ is at least distance $\frac{c}{\log (T + 2)}$ away from any zero or pole

Then

| \frac{ζ^{'} (s)}{ζ (s)} | ≪_{c} {(\log (T + 2))}^{2} .