Fourth moments of prime exponential sums and exceptional sets for sums and differences of primes
Year: 2026
Type: preprint
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.
Source: arXiv (Cornell University)
Authors: Li, Huixi, Liu, Zhixin
Institutions: Nankai University, Tianjin University
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Award: G8974978213
OpenAlex ID: W7223002205