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Explicit lower bounds for residues at \(s=1\) of Dedekind zeta functions and relative class numbers of CM-fields. (English) Zbl 1026.11085
This article deals with obtaining explicit lower bounds for residues \(\kappa_K\) at \(s = 1\) of Dedekind zeta functions of number fields \(K\), and gives various applications.
In the first theorem, lower bounds for \(\kappa_K\) are given, e.g. if \(K\) is a totally complex number field with large enough degree and root discriminant, and if \(\zeta_K(\beta) \leq 0\) for some \(\beta \in [1 - (2/\log \operatorname {disc} K),1)\). Combined with upper bounds on \(\kappa_k\) for the maximal real subfield \(k\) of the CM-field \(K\), lower bounds for the relative class number \(h_K^-\) are obtained.
These lower bounds on \(h_K^-\) are then used to prove a result of Brauer-Siegel type: for a sequence of CM fields whose root discriminants tend to infinity, \(\log h_K^-\) is asymptotically equal to \(\frac 12 \log (\operatorname {disc}K/\operatorname {disc}k)\).
Finally, the author determines lower bounds for relative class numbers of nonnormal quartic CM-fields that were used for determining all nonnormal quartic and dihedral octic CM-fields with class groups of exponent \(\leq 2\).

MSC:
11R42 Zeta functions and \(L\)-functions of number fields
11R29 Class numbers, class groups, discriminants
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