  In this paper, we prove an unconditional form of Fujii's formula for the
average number of Goldbach representations and show that the error in this
formula is determined by a general zero-free region of the Riemann
zeta-function, and vice versa. In particular, we describe the error in the
unconditional formula in terms of the remainder in the Prime Number Theorem
which connects the error to zero-free regions of the Riemann zeta-function.
