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A discrete mean-value theorem for the higher derivatives of the Riemann zeta function
[Submitted on 6 Jun 2021 (v1), last revised 22 May 2026 (this ve · 2026-05-25 · via math updates on arXiv.org

Mathematics > Number Theory

arXiv:2106.03005 (math)

[Submitted on 6 Jun 2021 (v1), last revised 22 May 2026 (this version, v3)]

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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.
Comments: This version adds a couple of improvements which were found after the paper was published (At the end of section 1, we state a better error term under RH and a neater way of presenting the main terms)
Subjects: Number Theory (math.NT)
MSC classes: 11M06, 11M26
Cite as: arXiv:2106.03005 [math.NT]
  (or arXiv:2106.03005v3 [math.NT] for this version)
  https://doi.org/10.48550/arXiv.2106.03005

arXiv-issued DOI via DataCite
Journal reference: Journal of Number Theory 241 (2022) 142--164
Related DOI: https://doi.org/10.1016/j.jnt.2022.03.004

DOI(s) linking to related resources

Submission history

From: Chris Hughes [view email]
[v1] Sun, 6 Jun 2021 01:59:38 UTC (317 KB)
[v2] Thu, 3 Mar 2022 15:12:29 UTC (319 KB)
[v3] Fri, 22 May 2026 17:45:23 UTC (319 KB)

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