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

Christopher Hughes; Andrew E B S 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 E B S Pearce-Crump

^*Corresponding author for this work

School of Mathematics

Research output: Contribution to journal › Article (Academic Journal) › peer-review

5 Citations (Scopus)

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	26 Apr 2022
DOIs	https://doi.org/10.1016/j.jnt.2022.03.004
Publication status	Published - 1 Dec 2022

Research Groups and Themes

Pure Mathematics

Access to Document

10.1016/j.jnt.2022.03.004Licence: CC BY

Handle.net

Persistent link

Cite this

@article{5891ae58654b4b9197ce9f658985f5c0,

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 Pearce-Crump, \{Andrew E B S\}",

year = "2022",

month = dec,

day = "1",

doi = "10.1016/j.jnt.2022.03.004",

language = "English",

volume = "241",

pages = "142--164",

journal = "Journal of Number Theory",

issn = "0022-314X",

publisher = "Academic Press Inc.",

}

TY - JOUR

T1 - A discrete mean-value theorem for the higher derivatives of the Riemann zeta function

AU - Hughes, Christopher

AU - Pearce-Crump, Andrew E B S

PY - 2022/12/1

Y1 - 2022/12/1

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.

U2 - 10.1016/j.jnt.2022.03.004

DO - 10.1016/j.jnt.2022.03.004

M3 - Article (Academic Journal)

SN - 0022-314X

VL - 241

SP - 142

EP - 164

JO - Journal of Number Theory

JF - Journal of Number Theory

ER -

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

Abstract

Research Groups and Themes

Access to Document

Handle.net

Fingerprint

Cite this