A discrete mean-value theorem for the higher derivatives of the Riemann zeta function
Number Theory
2026-05-25 v3
Abstract
We show that the 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 odd/even, respectively. We show this by giving a full asymptotic expansion of these sums.
Keywords
Cite
@article{arxiv.2106.03005,
title = {A discrete mean-value theorem for the higher derivatives of the Riemann zeta function},
author = {Christopher Hughes and Andrew Pearce-Crump},
journal= {arXiv preprint arXiv:2106.03005},
year = {2026}
}
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)