Logarithmic derivative inside the zeta zero-free region
= Logarithmic derivative inside the zeta zero-free region
After shrinking the zero-free-region constant, one has
$$
\frac{\zeta'(\sigma+it)}{\zeta(\sigma+it)}\ll\log|t|
$$
in the half-width region $\sigma>1-c/(2\log|t|)$ for $|t|$ large. Compare the local partial-fraction expansion at $s$ with one at a point just to the right of one, where the <Euler product> controls the logarithmic derivative.