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Let $R(n) = \sum_{a+b=n} \Lambda(a)\Lambda(b)$, where $\Lambda(\cdot)$ is thevon Mangoldt function. The function $R(n)$ is often studied in connection withGoldbach's conjecture. On the Riemann hypothesis (RH) it is known that$\sum_{n\leq x} R(n) = x^2/2 - 4x^{3/2} G(x) + O(x^{1+\epsilon})$, where$G(x)=\Re \sum_{\gamma>0} \frac{x^{i\gamma}}{(\frac{1}{2} +i\gamma)(\frac{3}{2} + i\gamma)}$ and the sum is over the ordinates of thenontrivial zeros of the Riemann zeta function in the upper half-plane. We prove(on RH) that each of the inequalities $G(x) < -0.02093$ and $G(x)> 0.02092$hold infinitely often, and establish improved bounds under an assumption oflinearly independence for zeros of the zeta function. We also show that thebounds we obtain are very close to optimal.

The size of oscillations in the Goldbach conjecture