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

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

Christopher Hughes, Andrew Pearce-Crump

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We show that the nth 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.

Original language	English
Pages (from-to)	142-164
Number of pages	23
Journal	Journal of Number Theory
Volume	241
Early online date	24 Aug 2022
DOIs	https://doi.org/10.1016/j.jnt.2022.03.004
Publication status	Published - Dec 2022

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10.1016/j.jnt.2022.03.004Licence: CC BY

Cite this

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title = "A discrete mean-value theorem for the higher derivatives of the Riemann zeta function",

abstract = "We show that the nth 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.",

author = "Christopher Hughes and Andrew Pearce-Crump",

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AU - Pearce-Crump, Andrew

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N2 - We show that the nth 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.

AB - We show that the nth 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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DO - 10.1016/j.jnt.2022.03.004

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JO - Journal of Number Theory

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