Low Lying Zeros of Dedekind Zeta Functions

Abstract

Let K be a number field of degree n_K, and let h(K) denote the smallest absolute value of the imaginary part of a non-trivial zero of its Dedekind zeta function. In 1979, Hoffstein proved the unconditional bound h(K)≤ 0.87+o(1) as n_K→∞. In a 1990 survey article, Odlyzko stated without proof that a different kernel in Weil's explicit formula could produce h(K)≤ 0.54+o(1). No further improvement on Hoffstein's $0.87$ threshold accompanied by a published proof has since appeared. We choose a linear kernel and first show that Hoffstein's original parameters already yield the improved constant $0.825$. We then generalise his linear kernel method using Stechkin's trick and prove the unconditional bound h(K)≤ 0.6+o(1) as n_K→∞, giving the best proved asymptotic bound to date.

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Cite

BibTeX RIS

BibTeX

@misc{trudgian2026lying,
  title = {{Low Lying Zeros of Dedekind Zeta Functions}},
  author = {Timothy S. Trudgian and Sebastian Zuniga-Alterman},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.16010},
  eprint = {2610.16010},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.16010},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Low Lying Zeros of Dedekind Zeta Functions
AU  - Timothy S. Trudgian
AU  - Sebastian Zuniga-Alterman
PY  - 2026
DA  - 2026/10/12
PB  - arXiv
DO  - 10.48550/arxiv.2610.16010
UR  - https://arxiv.org/abs/2610.16010
AB  - Let K be a number field of degree n_K, and let h(K) denote the smallest absolute value of the imaginary part of a non-trivial zero of its Dedekind zeta function. In 1979, Hoffstein proved the unconditional bound h(K)≤ 0.87+o(1) as n_K→∞. In a 1990 survey article, Odlyzko stated without proof that a different kernel in Weil's explicit formula could produce h(K)≤ 0.54+o(1). No further improvement on Hoffstein's $0.87$ threshold accompanied by a published proof has since appeared. We choose a linear kernel and first show that Hoffstein's original parameters already yield the improved constant $0.825$. We then generalise his linear kernel method using Stechkin's trick and prove the unconditional bound h(K)≤ 0.6+o(1) as n_K→∞, giving the best proved asymptotic bound to date.
ER  -

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