Explicit Lower bounds for residues at of Dedekind zeta functions and relative class numbers of CM-fields |
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Authors: | Sté phane Louboutin |
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Affiliation: | Institut de Mathématiques de Luminy, UPR 9016, 163 avenue de Luminy, Case 907, 13288 Marseille Cedex 9, France |
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Abstract: | Let be a given set of positive rational primes. Assume that the value of the Dedekind zeta function of a number field is less than or equal to zero at some real point in the range . We give explicit lower bounds on the residue at of this Dedekind zeta function which depend on , the absolute value of the discriminant of and the behavior in of the rational primes . Now, let be a real abelian number field and let be any real zero of the zeta function of . We give an upper bound on the residue at of which depends on , and the behavior in of the rational primes . By combining these two results, we obtain lower bounds for the relative class numbers of some normal CM-fields which depend on the behavior in of the rational primes . We will then show that these new lower bounds for relative class numbers are of paramount importance for solving, for example, the exponent-two class group problem for the non-normal quartic CM-fields. Finally, we will prove Brauer-Siegel-like results about the asymptotic behavior of relative class numbers of CM-fields. |
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Keywords: | Dedekind zeta functions CM-field relative class number |
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