A HARMONIC SUM over NONTRIVIAL ZEROS of the RIEMANN ZETA-FUNCTION

Richard P. Brent; David J. Platt; Timothy S. Trudgian

doi:10.1017/S0004972720001252

A HARMONIC SUM over NONTRIVIAL ZEROS of the RIEMANN ZETA-FUNCTION

Richard P. Brent
, David J. Platt
, Timothy S. Trudgian

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

2 Citations (Scopus)

Abstract

We consider the sum, where ranges over the ordinates of nontrivial zeros of the Riemann zeta-function in an interval, and examine its behaviour as. We show that, after subtracting a smooth approximation the sum tends to a limit, which can be expressed as an integral. We calculate H to high accuracy, using a method which has error. Our results improve on earlier results by Hassani ['Explicit approximation of the sums over the imaginary part of the non-trivial zeros of the Riemann zeta function', Appl. Math. E-Notes 16 (2016), 109-116] and other authors.

Original language	English
Pages (from-to)	59-65
Number of pages	7
Journal	Bulletin of the Australian Mathematical Society
Volume	104
Issue number	1
DOIs	https://doi.org/10.1017/S0004972720001252
Publication status	Published - Aug 2021

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title = "A HARMONIC SUM over NONTRIVIAL ZEROS of the RIEMANN ZETA-FUNCTION",

abstract = "We consider the sum, where ranges over the ordinates of nontrivial zeros of the Riemann zeta-function in an interval, and examine its behaviour as. We show that, after subtracting a smooth approximation the sum tends to a limit, which can be expressed as an integral. We calculate H to high accuracy, using a method which has error. Our results improve on earlier results by Hassani ['Explicit approximation of the sums over the imaginary part of the non-trivial zeros of the Riemann zeta function', Appl. Math. E-Notes 16 (2016), 109-116] and other authors.",

keywords = "Riemann zeta-function, acceleration, nontrivial zeros",

author = "Brent, \{Richard P.\} and Platt, \{David J.\} and Trudgian, \{Timothy S.\}",

note = "Publisher Copyright: {\textcopyright} 2020 Australian Mathematical Publishing Association Inc..",

year = "2021",

month = aug,

doi = "10.1017/S0004972720001252",

language = "English",

volume = "104",

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AU - Brent, Richard P.

AU - Platt, David J.

AU - Trudgian, Timothy S.

PY - 2021/8

Y1 - 2021/8

N2 - We consider the sum, where ranges over the ordinates of nontrivial zeros of the Riemann zeta-function in an interval, and examine its behaviour as. We show that, after subtracting a smooth approximation the sum tends to a limit, which can be expressed as an integral. We calculate H to high accuracy, using a method which has error. Our results improve on earlier results by Hassani ['Explicit approximation of the sums over the imaginary part of the non-trivial zeros of the Riemann zeta function', Appl. Math. E-Notes 16 (2016), 109-116] and other authors.

AB - We consider the sum, where ranges over the ordinates of nontrivial zeros of the Riemann zeta-function in an interval, and examine its behaviour as. We show that, after subtracting a smooth approximation the sum tends to a limit, which can be expressed as an integral. We calculate H to high accuracy, using a method which has error. Our results improve on earlier results by Hassani ['Explicit approximation of the sums over the imaginary part of the non-trivial zeros of the Riemann zeta function', Appl. Math. E-Notes 16 (2016), 109-116] and other authors.

KW - Riemann zeta-function

KW - acceleration

KW - nontrivial zeros

UR - https://www.scopus.com/pages/publications/85096680798

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DO - 10.1017/S0004972720001252

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EP - 65

JO - Bulletin of the Australian Mathematical Society

JF - Bulletin of the Australian Mathematical Society

IS - 1

ER -

A HARMONIC SUM over NONTRIVIAL ZEROS of the RIEMANN ZETA-FUNCTION

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