A discrete mean-value theorem for the higher derivatives of the Riemann zeta function

Christopher Hughes; Andrew Pearce-Crump

doi:10.1016/j.jnt.2022.03.004

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arXiv · 2106.03005

A discrete mean-value theorem for the higher derivatives of the Riemann zeta function

Christopher Hughes·Andrew Pearce-Crump

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Abstract

We show that the $n$th derivative of the Riemann zeta function, when summed over the non-trivial zeros of zeta, is real and positive/negative in the mean for $n$ odd/even, respectively. We show this by giving a full asymptotic expansion of these sums.

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Christopher Hughes, Andrew Pearce-Crump. 2021-06-06. A discrete mean-value theorem for the higher derivatives of the Riemann zeta function. https://doi.org/10.1016/j.jnt.2022.03.004

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