Explicit smoothed prime ideals theorems under GRH

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