Skip to content
DraconiticResearch
Search Sources API
201,683 studies31 sourcesUpdated 59 min ago
PreprintarXiv licencearXiv

Upper bounds for arithmetic functions over polynomial values via Chebotarev--Sato--Tate distributions

  • Jiong Yang
Open at sourcePDFDOICite

Abstract

We establish logarithmic upper bounds for nonnegative multiplicative functions evaluated at values of multivariable polynomials. The polynomial contribution is encoded by the permutation character on the geometric irreducible components of the corresponding hypersurface, while the arithmetic contribution is described by a class function on a joint Chebotarev--Sato--Tate group. This yields a unified framework for symmetric-power coefficients of CM and non-CM modular forms, Dedekind zeta-function coefficients, and mixed automorphic--Galois weights. As a further application, we obtain quantitative divisibility results for Fourier coefficients along polynomial values, including explicit formulas in the full residual-image and Eisenstein congruence cases.

Related

  1. Arithmetic k-Polynomial Dynamical Localization for k Power-Law Quasi-Periodic Long-Range Operators on ℓ2(ℤd)
    Ao Cai, Huihui Lv, Yuan Shan +1
    arXivPreprintarXiv licenceFull textDynamical systemsarXiv
  2. Congruence Jumping in Polynomial Divisibility Systems: Arithmetic Structure and Solution Sets
    Max A. Alekseyev, Dmitry I. Khomovsky
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv
  3. Triple divisor type functions in arithmetic progressions to prime power moduli
    Hui wang, Tengyou Zhu
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryarXiv
  4. Truncated pretentious distances of multiplicative arithmetic functions
    Thomas Renard
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv
  5. The Arithmetic of Spectra: Factorization, Statistics, and Symmetric Functions
    A. Chaudhary
    arXivPreprintCC BY 4.0Full textMathematical physicsStatistical & nonlinear physicsarXiv

Cite

BibTeXRIS
BibTeX
@misc{yang2026upper,
  title = {{Upper bounds for arithmetic functions over polynomial values via Chebotarev--Sato--Tate distributions}},
  author = {Jiong Yang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.07080},
  eprint = {2609.07080},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.07080},
}
RIS
TY  - UNPB
TI  - Upper bounds for arithmetic functions over polynomial values via Chebotarev--Sato--Tate distributions
AU  - Jiong Yang
PY  - 2026
DA  - 2026/09/07
PB  - arXiv
DO  - 10.48550/arxiv.2609.07080
UR  - https://arxiv.org/abs/2609.07080
AB  - We establish logarithmic upper bounds for nonnegative multiplicative functions evaluated at values of multivariable polynomials. The polynomial contribution is encoded by the permutation character on the geometric irreducible components of the corresponding hypersurface, while the arithmetic contribution is described by a class function on a joint Chebotarev--Sato--Tate group. This yields a unified framework for symmetric-power coefficients of CM and non-CM modular forms, Dedekind zeta-function coefficients, and mixed automorphic--Galois weights. As a further application, we obtain quantitative divisibility results for Fourier coefficients along polynomial values, including explicit formulas in the full residual-image and Eisenstein congruence cases.
ER  -