Explicit smoothed prime ideals theorems under GRH

Let ψK be the Chebyshev function of a number field K. Let ψ(1) K (x) := x 0 ψK(t) dt and ψ(2) K (x) := 2 x 0 ψ(1) K (t) dt. We prove under GRH (Generalized Riemann Hypothesis) explicit inequalities for the differences |ψ(1) K (x)− x2 2 | and |ψ(2) K (x)− x3 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.