Global partial-fraction expansion of the zeta logarithmic derivative ($-{\zeta }^{\prime }/\zeta (s)=\frac{1}{s-1}-\frac{1}{2}\log \pi +\frac{1}{2}\psi (1+s/2)-B-{\sum }_{\rho }(\frac{1}{s-\rho }+\frac{1}{\rho })$)

Here $\psi ={\Gamma }^{\prime }/\Gamma$ is the digamma function, $B=\log 2+\frac{1}{2}\log \pi -1-\gamma /2$, and $\gamma$ is the Euler--Mascheroni constant. The sum runs over Nontrivial zeros of the Riemann zeta function with multiplicities. Each displayed paired summand is $O_s(|\rho {|}^{-2})$, so it converges locally uniformly away from zeros. Differentiate the genus-one Hadamard factorization of the Riemann xi function and use the Gamma function recurrence. Unlike an unpaired reciprocal-zero sum, this expression does not require an unstated summation convention.