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Write $s=\sigma +it$ fixed, with $\xi (s)\ne 0$. The zeros satisfy $0<\beta <1$. For $|\gamma |>2(|t|+1)$,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 $O(j{2}^{-j})$. 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 $1/(s-\rho )$.

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.