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

Shanks conjectured that $\zeta ' (\rho)$, 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 and its proof, 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 $\zeta^{(n)}(\rho)$.