Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 23 / 3 / b / Solution

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 $1/(s-\rho )$.