Abstract
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 ζ K ( 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.
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