Riemann xi function ($\xi (s)=\frac{1}{2}s(s-1){\pi }^{-s/2}\Gamma (s/2)\zeta (s)$)

Removing the completion's poles produces an entire function with $\xi (0)=\xi (1)=1/2$ and $\xi (s)=\xi (1-s)$. Its zeros are the nontrivial zeros of the Riemann zeta function. Its finite order allows Hadamard factorization, while the xi modulus criterion for the Riemann hypothesis characterizes their horizontal location.

Differentiate the genus-one Hadamard factorization. Its complex difference terms converge as $O(|\rho {|}^{-2})$, and the individual real sums converge by the absolute convergence of the real xi logarithmic derivative. The remaining constant is $\Re B+\sum \Re (1/\rho )$. On the critical line the xi logarithmic derivative has zero real part, while the kernel terms cancel in pairs reflected across that line. Hence this constant vanishes and gives the displayed formula away from zeros.

The Jensen zero-count bound gives $O(T\log T)$ zeros up to ordinate $T$. Since their real parts lie in $(0,1)$, a fixed-point real logarithmic derivative summand is $O(|\Im \rho {|}^{-2})$. Dyadic ordinate bands then contribute $O(j{2}^{-j})$. This proves absolute convergence of the real sum, while the unpaired complex sum of reciprocals need not converge absolutely.