Abstract
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.
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Louboutin, S.R. Some explicit upper bounds for residues of zeta functions of number fields taking into account the behavior of the prime 2. manuscripta math. 125, 43–67 (2008). https://doi.org/10.1007/s00229-007-0132-0
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DOI: https://doi.org/10.1007/s00229-007-0132-0