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

Shanks conjectured that ζ ′ (ρ), where ρ 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 generalization to all higher-order derivatives ζ ⁽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 ζ ⁽ⁿ⁾ (ρ).