A discrete mean value of the derivative of the Riemann zeta function

In this article we compute a discrete mean value of the derivative of the Riemann zeta function. This mean value will be important for several applications concerning the size of ${\zeta }^{\mathrm{\prime }}(\rho )$ where $\zeta (s)$ is the Riemann zeta function and $\rho$ is a non-trivial zero of the Riemann zeta function.