Probabilistic properties of number fields

In general a bound on number theoretic invariants under the Generalized Riemann Hypothesis (GRH) for the Dedekind zeta function of a number field K is much stronger than an unconditional one. In this article, we consider three invariants; the residue of ζK(s) at s=1, the logarithmic derivative of Artin L-function attached to K at s=1, and the smallest prime which does not split completely in K. We obtain bounds on them just as good as the bounds under GRH except for a density zero set of number fields.