Explicit smoothed prime ideals theorems under GRH

Abstract:

Let $\psi _{\mathbb {K}}$ be the Chebyshev function of a number field $\mathbb {K}$. Let $\psi ^{(1)}_{\mathbb {K}}(x):=\int _{0}^{x}\psi _{\mathbb {K}}(t) \mathrm {d} t$ 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 $|\psi ^{(1)}_{\mathbb {K}}(x) - \tfrac {x^2}{2}|$ 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.