  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.
