= Solution
Write $s=\sigma+it$ fixed, with $\xi(s)\ne0$. The zeros satisfy $0<\beta<1$. For $|\gamma|>2(|t|+1)$,
$$
\left|\frac{\sigma-\beta}{(\sigma-\beta)^2+(t-\gamma)^2}\right|
\le\frac{4(|\sigma|+1)}{\gamma^2}.
$$
The previous bound gives at most $O(2^j j)$ zeros in each dyadic ordinate band $2^j\le|\gamma|<2^{j+1}$, so its total majorant is $O(j2^{-j})$. That series converges. There are only finitely many zeros in the remaining bounded bands, and none has the forbidden denominator zero at the specified nonzero point of $\xi$. Thus the real <logarithmic derivative> sum converges absolutely. This proves the <absolute convergence of the real xi logarithmic derivative> without claiming <absolute convergence> of the unpaired complex sums of $1/(s-\rho)$.
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