On the discrete mean of the derivative of Hardy's $Z$-function

Abstract

Update: This result was obtained by Milinovich with a better error term. He used $\zeta'(s)$, but we considered $Z'(t)$. We corrected a typo in the main theorem. We consider the sum of the square of the derivative of Hardy's $Z$-function over the zeros of Hardy's $Z$-function. If the Riemann Hypothesis is true, it is equal to the sum of $|\zeta'(\rho)|^2$, where $\rho$ runs over the zeros of the Riemann zeta-function. In 1984, Gonek obtained an asymptotic formula for the sum. In this paper we prove a sharper formula.