A singular series average and the zeros of the Riemann zeta-function

We show that the Riesz mean of the singular series in the Goldbach and the Hardy-Littlewood prime-pair conjectures has an asymptotic formula with an error term that can be expressed as an explicit formula that depends on the zeros of the Riemann zeta-function. Unconditionally this error term can be...

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Bibliographic Details
Published in:	arXiv.org
Main Authors:	Goldston, D A, Ade, Irma Suriajaya
Format:	Paper
Language:	English
Published:	Ithaca Cornell University Library, arXiv.org 31.07.2020
ISSN:	2331-8422
Online Access:	Get full text
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Summary:	We show that the Riesz mean of the singular series in the Goldbach and the Hardy-Littlewood prime-pair conjectures has an asymptotic formula with an error term that can be expressed as an explicit formula that depends on the zeros of the Riemann zeta-function. Unconditionally this error term can be shown to oscillate, while conditionally it can be shown to oscillate between sharp bounds.
Bibliography:	SourceType-Working Papers-1 ObjectType-Working Paper/Pre-Print-1 content type line 50
ISSN:	2331-8422
DOI:	10.48550/arxiv.2007.16099