Local partial-fraction expansion of the Riemann zeta logarithmic derivative

Uniformly for $s=\sigma +it$ with $1/2\le \sigma \le 2$ and $|t|\ge 3$, away from zeros,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.