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arXiv · 2610.26756

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

Huixi Li·Zhixin Liu
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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 $1/2\leqΘ<11/12$, then $E_+(N)\ll_ΘN^{1/2}(\log N)^{100}$ and $E_-(N)\ll_ΘN^{1/2}(\log N)^{100}$. 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.

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Huixi Li, Zhixin Liu. 2026-10-19. Fourth moments of prime exponential sums and exceptional sets for sums and differences of primes. https://arxiv.org/abs/2610.26756

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