Fourth moments of prime exponential sums and exceptional sets for sums and differences of primes

Abstract: We prove a square root bound for the exceptional sets in the binary Goldbach and prime difference problems under quasi Riemann Hypothesis. If every nontrivial zero of every primitive Dirichlet $L$-function has real part at most a fixed $1/2\leqΘ&lt;11/12$, then $E_+(N)\ll_ΘN^{1/2}(\log N)<sup>{100}$ and $E_-(N)\ll_ΘN^{1/2}(\log N)<sup>{100}$. Here $E_+(N)$ counts the even integers greater than $2$ and at most $N$ that are not sums of two primes, and $E_-(N)$ counts the positive even integers at most $N$ that are not differences of two primes. The common analytic estimate is a fourth moment bound of the minor arcs for a smoothed prime exponential sum.