The average number of Goldbach representations and Zero-free regions of the Riemann zeta function

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.