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PreprintCC BY 4.0arXiv

A GL3 level analog of Selberg's result for S(t)

  • Qingfeng Sun
  • Hui Wang
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Abstract

Let π be a spherical cuspidal GL3 newform of prime level N and trivial central character, and define S(t,π)=π−1arg L(1/2+it,π). Assuming the generalized Riemann hypothesis for L(s,π), we establish an asymptotic formula for each fixed weighted moment of S(t,π) as N→∞, for fixed t>0. Consequently, under the same weights, S(t,π)/√(loglog N) converges in distribution to a centered Gaussian distribution of variance 1/(2π2). The key input is a weighted orthogonality formula derived from the Kuznetsov formula of prime level.

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Cite

BibTeXRIS
BibTeX
@misc{sun2026level,
  title = {{A $\mathrmGL_3$ level analog of Selberg's result for $S(t)$}},
  author = {Qingfeng Sun and Hui Wang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.23201},
  eprint = {2610.23201},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.23201},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - A GL_3 level analog of Selberg's result for S(t)
AU  - Qingfeng Sun
AU  - Hui Wang
PY  - 2026
DA  - 2026/10/16
PB  - arXiv
DO  - 10.48550/arxiv.2610.23201
UR  - https://arxiv.org/abs/2610.23201
AB  - Let π be a spherical cuspidal GL_3 newform of prime level N and trivial central character, and define S(t,π)=π^−1arg L(1/2+it,π). Assuming the generalized Riemann hypothesis for L(s,π), we establish an asymptotic formula for each fixed weighted moment of S(t,π) as N→∞, for fixed t>0. Consequently, under the same weights, S(t,π)/√(loglog N) converges in distribution to a centered Gaussian distribution of variance 1/(2π^2). The key input is a weighted orthogonality formula derived from the Kuznetsov formula of prime level.
ER  -