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

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