A heuristic for discrete mean values of the derivative of the Riemann zeta function

Shanks conjectured that ${\zeta }^{\mathrm{\prime }}(\rho )$, where $\rho$ ranges over non-trivial zeros of the Riemann zeta function, is real and positive in the mean. We present a history of this problem, including a generalisation to all higher-order derivatives ${\zeta }^{(n)}(s)$, for which the sign of the mean alternatives between positive for odd $n$ and negative for even $n$. Furthermore, we give a simple heuristic that provides the leading term (including its sign) of the asymptotic formula for the average value of ${\zeta }^{(n)}(\rho )$.