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

Abstract

Shanks conjectured that $\hat{I}{\P }^{\prime }(\ddot{I})$, where $\ddot{I}$ 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 $\hat{I}{\P }^{(n)}(s)$, 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 $\hat{I}{\P }^{(n)}(\ddot{I})$.

To appear in the journal Integers