The size of oscillations in the Goldbach conjecture

TL;DR

This paper investigates the oscillations in the function related to Goldbach's conjecture, proving that certain bounds on these oscillations occur infinitely often under the Riemann hypothesis and related assumptions.

Contribution

It establishes that the oscillation function exceeds specific bounds infinitely often on RH and improves these bounds assuming linear independence of zeta zeros.

Findings

Proves oscillation bounds occur infinitely often under RH.

Provides improved bounds assuming linear independence of zeta zeros.

Shows bounds are close to optimal.

Abstract

Let $R(n) = \sum_{a+b=n} \Lambda(a)\Lambda(b)$, where $\Lambda(\cdot)$ 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 $\sum_{n\leq x} R(n) = x^2/2 - 4x^{3/2} G(x) + O(x^{1+\epsilon})$, 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 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.