paper

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

arXiv:2106.03005 · doi:10.1016/j.jnt.2022.03.004

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.

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)

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