Explicit smoothed prime ideals theorems under GRH

Let $\psi _{\mathbb{K}}$ be the Chebyshev function of a number field $\mathbb{K}$. Let $\psi ^{(1)}_{\mathbb{K}}(x):=\int _{0}^{x}\psi _{\mathbb{K}}(t)\,\mathrm {d} t$ and $\psi ^{(2)}_{\mathbb{K}}(x):=2\int _{0}^{x}\psi ^{(1)}_{\mathbb{K}}(t)\,\mathrm {d} t$. We prove under GRH (Generalized Riemann Hypothesis) explicit inequalities for the differences
$\vert\psi ^{(1)}_{\mathbb{K}}(x) - \tfrac {x^2}{2}\vert$ and $\vert\psi ^{(2)}_{\mathbb{K}}(x) - \tfrac {x^3}{3}\vert$. We deduce an efficient algorithm for the computation of the residue of the Dedekind zeta function and a bound on small-norm prime ideals.