Some explicit upper bounds for residues of zeta functions of number fields taking into account the behavior of the prime 2

We recall the known explicit upper bounds for the residue at s = 1 of the Dedekind zeta function of a number field K. Then, we improve upon these previously known upper bounds by taking into account the behavior of the prime 2 in K. We finally give several examples showing how such improvements yield better bounds on the absolute values of the discriminants of CM-fields of a given relative class number. In particular, we will obtain a 4,000-fold improvement on our previous bound for the absolute values of the discriminants of the non-normal sextic CM-fields with relative class number one.