A singular series average and the zeros of the Riemann zeta-function
Volume 200 / 2021
Acta Arithmetica 200 (2021), 71-90
MSC: 11N05, 11M26.
DOI: 10.4064/aa200821-24-2
Published online: 28 June 2021
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.
