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arXiv · 2610.14257

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 $κ_{\mathbb{K}}$ be the residue at $1$ of the Dedekind zeta function of a number field $\mathbb{K}$. We prove explicit upper and lower bounds for $κ_{\mathbb{K}}$ 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 $\leq 2000$ 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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Stephan Ramon Garcia, Loïc Grenié, Ethan Simpson Lee, Giuseppe Molteni. 2026-10-09. Bounds for the residue of Dedekind zeta functions under GRH and applications. https://arxiv.org/abs/2610.14257

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