The size of oscillations in the Goldbach conjecture

Abstract

Let R(n) = a+b=n Λ(a)Λ(b), where Λ(•) is the von Mangoldt function. The function R(n) is often studied in connection with Goldbach's conjecture. On the Riemann hypothesisand 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) < −0.02093 and G(x) > 0.02092 hold infinitely often, and establish improved bounds 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.