= Global partial-fraction expansion of the zeta logarithmic derivative
{title2=$-\zeta'/\zeta(s)=\frac1{s-1}-\frac12\log\pi+\frac12\psi(1+s/2)-B-\sum_\rho(\frac1{s-\rho}+\frac1\rho)$}
Here $\psi=\Gamma'/\Gamma$ is the <digamma function>, $B=\log2+\frac12\log\pi-1-\gamma/2$, 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.
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