Skip to content
DraconiticResearch
Search Sources API
181,716 studies31 sourcesUpdated 20 min ago
PreprintarXiv licencearXiv

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

  • Huixi Li
  • Zhixin Liu
Open at sourcePDFDOICite

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≤Θ<11/12, then E+(N)≪ΘN1/2(log N)100 and E-(N)≪ΘN1/2(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.

Related

  1. Spectral fourth moments of Hecke--Maaß cusp forms
    Edgar Assing, Peter Humphries
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv
  2. On Hecke algebras at exceptional Eisenstein primes
    Alexandre Maksoud
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryarXiv
  3. A Bombieri-Vinogradov theorem for exponential sums over products of k primes
    Péa Bazin
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryarXiv
  4. Order Reduction of Exponential Runge--Kutta Methods: Fourth-Order Schemes for Non-Commuting Operators
    Thi Tam Dang, Pablo Alexei Gazca-Orozco, Trung Hau Hoang
    arXivPreprintCC BY-NC-ND 4.0Full textNumerical analysis & scientific computingarXiv
  5. Exponential decay of fractional moments in the H2|2 model on ℤ2
    Jinglin Wang, Xiaolin Zeng
    arXivPreprintCC BY-NC-ND 4.0Full textMathematical physicsProbabilityarXiv

Cite

BibTeXRIS
BibTeX
@misc{li2026fourth,
  title = {{Fourth moments of prime exponential sums and exceptional sets for sums and differences of primes}},
  author = {Huixi Li and Zhixin Liu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.26756},
  eprint = {2610.26756},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.26756},
}
RIS
TY  - UNPB
TI  - Fourth moments of prime exponential sums and exceptional sets for sums and differences of primes
AU  - Huixi Li
AU  - Zhixin Liu
PY  - 2026
DA  - 2026/10/19
PB  - arXiv
DO  - 10.48550/arxiv.2610.26756
UR  - https://arxiv.org/abs/2610.26756
AB  - 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≤Θ<11/12, then E_+(N)≪_ΘN^1/2(log N)^100 and E_-(N)≪_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.
ER  -