= Local partial-fraction expansion of the Riemann zeta logarithmic derivative
{c}
Uniformly for $s=\sigma+it$ with $1/2\le\sigma\le2$ and $|t|\ge3$, away from zeros,
$$
\frac{\zeta'(s)}{\zeta(s)}=\sum_{\rho:\,|\operatorname{Im}\rho-t|\le1}\frac1{s-\rho}+O(\log(|t|+3)).
$$
The sum runs over <Nontrivial zeros of the Riemann zeta function>, with <multiplicity>. The height condition keeps this strip away from the <pole> at $s=1$ and the <trivial zeros of the Riemann zeta function>; these cannot simply be omitted in an arbitrary disk.
Start with the <global partial-fraction expansion of the zeta logarithmic derivative> and compare it at $s$ and $2+it$. The <Euler product> bounds the <logarithmic derivative> at $2+it$. The nearby zero terms remain explicit, while the paired differences from distant height bands are bounded using the <local zero count for the Riemann zeta function> and a convergent inverse-square tail. The <gamma function> contribution is also logarithmic by the <Stirling formula>. This yields the stated local error without an undefined convention for which zeros are nearby.
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