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arXiv · 2305.14253

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

Christopher Hughes·Greg Martin·Andrew Pearce-Crump
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Abstract

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, 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 $\zeta^{(n)}(\rho)$.

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Christopher Hughes, Greg Martin, Andrew Pearce-Crump. 2023-05-23. A heuristic for discrete mean values of the derivative of the Riemann zeta function. https://arxiv.org/abs/2305.14253

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