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

Source: [A singular series average and the zeros of the Riemann zeta-function](<https://www.sota2.com/research/paper/68fa587b1e209131ce9aebab>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2025-10-25T15:22:37.449Z

Authors: D. A. Goldston, Ade Irma Suriajaya

Published: 2020-07-31T14:24:55.000Z

## Abstract

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.

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## Original sources and implementation

[Original paper](<https://arxiv.org/abs/2007.16099>)

[PDF](<https://arxiv.org/pdf/2007.16099.pdf>)