Explicit smoothed prime ideals theorems under GRH

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