Recent results about the prime ideal theorem

Let ψ K ψK be the Chebyshev function of a number field K K , and let ψ (1) K (x):=∫ x 0 ψ K (t)dt ψK(1)(x):=∫0xψK(t)dt and ψ (2) K (x):=2∫ x 0 ψ (1) K (t)dt ψK(2)(x):=2∫0xψK(1)(t)dt. Assuming the truth of Riemann’s hypothesis for Dedekind’s zeta function of K K it is possible to prove explicit bounds for |ψ K (x)−x| |ψK(x)−x| , |ψ (1) K (x)−x 2 2 | |ψK(1)(x)−x22| and |ψ (2) K (x)−x 3 3 | |ψK(2)(x)−x33|. These results can be used to prove the existence of many ideals with a small norms and to produce an extremely efficient algorithm for the computation of the residue of Dedekind’s zeta function.