Absolute convergence of the real xi logarithmic derivative
= Absolute convergence of the real xi logarithmic derivative
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 $O(j2^{-j})$. This proves <absolute convergence> of the real sum, while the unpaired complex sum of reciprocals need not converge absolutely.