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PreprintCC0arXiv

An asymptotic Jensen-Polya criterion for GRH

  • Ken Ono
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Abstract

The Jensen--Pólya criterion expresses the generalized Riemann hypothesis (GRH) for Dedekind zeta-functions as a real-rootedness condition on their Jensen polynomials. Previous asymptotic results require large shifts, while computations establish real-rootedness in the Riemann case through a finite range of degrees. Here we study the unshifted family as the degree tends to infinity. For every number field K, we prove unconditionally that JK,d(X) is nonzero and has at least d−CK√(dlog d) distinct negative real zeros for all sufficiently large d, where CK>0 depends only on K. Therefore, the proportion of real zeros tends to one with error OK(√(log d/d)). The main theorems have been autoformalized in Lean~4 by AxiomProver.

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Cite

BibTeXRIS
BibTeX
@misc{ono2026asymptotic,
  title = {{An asymptotic Jensen-Polya criterion for GRH}},
  author = {Ken Ono},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.38870},
  eprint = {2610.38870},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.38870},
  note = {CC0},
}
RIS
TY  - UNPB
TI  - An asymptotic Jensen-Polya criterion for GRH
AU  - Ken Ono
PY  - 2026
DA  - 2026/10/28
PB  - arXiv
DO  - 10.48550/arxiv.2610.38870
UR  - https://arxiv.org/abs/2610.38870
AB  - The Jensen--Pólya criterion expresses the generalized Riemann hypothesis (GRH) for Dedekind zeta-functions as a real-rootedness condition on their Jensen polynomials. Previous asymptotic results require large shifts, while computations establish real-rootedness in the Riemann case through a finite range of degrees. Here we study the unshifted family as the degree tends to infinity. For every number field K, we prove unconditionally that J_K,d(X) is nonzero and has at least d−C_K√(dlog d) distinct negative real zeros for all sufficiently large d, where C_K>0 depends only on K. Therefore, the proportion of real zeros tends to one with error O_K(√(log d/d)). The main theorems have been autoformalized in Lean~4 by AxiomProver.
ER  -