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 |
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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