Explicit smoothed prime ideals theorems under GRH

Abstract

Let $\psi_{\mathbb K}$ be the Chebyshev function of a number field $\mathbb K$. Let $\psi^{(1)}_{\mathbb K}(x):=\int_{0}^{x}\psi_{\mathbb K}(t)\,d t$ 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.