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

TL;DR

This paper links the behavior of the singular series average in prime conjectures to the zeros of the Riemann zeta-function, providing explicit formulas and oscillation results for the error term.

Contribution

It establishes an explicit formula for the error term in the singular series average, connecting it to the zeros of the Riemann zeta-function, with unconditional and conditional oscillation results.

Findings

Error term oscillates unconditionally

Error term oscillates within sharp bounds conditionally

Explicit formula depends on the zeros of the Riemann zeta-function

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.