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Bounds for the residue of Dedekind zeta functions under GRH and applications

  • Stephan Ramon Garcia
  • Loïc Grenié
  • Ethan Simpson Lee
  • Giuseppe Molteni
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Abstract

Let κ𝕂 be the residue at $1$ of the Dedekind zeta function of a number field 𝕂. We prove explicit upper and lower bounds for κ𝕂 under the Generalized Riemann Hypothesis (GRH). Previous explicit bounds of comparable generality are weaker and require a weak form of Artin's holomorphy conjecture: our bounds improve the constants while assuming only GRH for the relevant Dedekind zeta functions. The new lower bound enables all imaginary quadratic fields with class number ≤ 2000 to be detected (assuming GRH). Furthermore, we demonstrate that the upper bound can be significantly reduced when considering extensions of any fixed, nontrivial field.

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Cite

BibTeXRIS
BibTeX
@misc{garcia2026bounds,
  title = {{Bounds for the residue of Dedekind zeta functions under GRH and applications}},
  author = {Stephan Ramon Garcia and Loïc Grenié and Ethan Simpson Lee and Giuseppe Molteni},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.14257},
  eprint = {2610.14257},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.14257},
}
RIS
TY  - UNPB
TI  - Bounds for the residue of Dedekind zeta functions under GRH and applications
AU  - Stephan Ramon Garcia
AU  - Loïc Grenié
AU  - Ethan Simpson Lee
AU  - Giuseppe Molteni
PY  - 2026
DA  - 2026/10/09
PB  - arXiv
DO  - 10.48550/arxiv.2610.14257
UR  - https://arxiv.org/abs/2610.14257
AB  - Let κ_𝕂 be the residue at $1$ of the Dedekind zeta function of a number field 𝕂. We prove explicit upper and lower bounds for κ_𝕂 under the Generalized Riemann Hypothesis (GRH). Previous explicit bounds of comparable generality are weaker and require a weak form of Artin's holomorphy conjecture: our bounds improve the constants while assuming only GRH for the relevant Dedekind zeta functions. The new lower bound enables all imaginary quadratic fields with class number ≤ 2000 to be detected (assuming GRH). Furthermore, we demonstrate that the upper bound can be significantly reduced when considering extensions of any fixed, nontrivial field.
ER  -