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

Let ${\kappa }_{\mathbb{K}}$ be the residue at $1$ of the Dedekind zeta function of a number field $\mathbb{K}$. We prove explicit upper and lower bounds for ${\kappa }_{\mathbb{K}}$ 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 $\le 2000$ to be detected (assuming GRH). Furthermore, we demonstrate that the upper bound can be significantly reduced when considering extensions of any fixed, nontrivial field.