Oscillations in the Goldbach conjecture

Let R(n) = Sigma(a+b= n) Lambda(a)Lambda(b), where Lambda( .) is the von Mangoldt function. The function R(n) is often studied in connection with Goldbach's conjecture. On the Riemann hypothesis (RH) it is known that Sigma(n <= x) R(n) = x(2)/2 - 4x(3/2)G(x) + O(x(1+is an element of) ), where G(x) = R Sigma(gamma>0) x(i gamma)/(1/2 + i gamma)(3/2 + i gamma) and the sum is over the ordinates of the nontrivial zeros of the Riemann zeta function in the upper half-plane. We prove (on RH) that each of the inequalities G(x) < -0.02297 and G(x) > 0.02103 holds infinitely often, and establish an improvement on the latter bound under an assumption of linearly independence for zeros of the zeta function. We also show that the bounds we obtain are very close to optimal.