Explicit Lower bounds for residues at 𝑠=1 of Dedekind zeta functions and relative class numbers of CM-fields

Abstract

Let S S be a given set of positive rational primes. Assume that the value of the Dedekind zeta function ζ K \zeta _K of a number field K K is less than or equal to zero at some real point β \beta in the range 1 2 > β > 1 {1\over 2} >\beta >1 . We give explicit lower bounds on the residue at s = 1 s=1 of this Dedekind zeta function which depend on β \beta , the absolute value d K d_K of the discriminant of K K and the behavior in K K of the rational primes p ∈ S p\in S . Now, let k k be a real abelian number field and let β \beta be any real zero of the zeta function of k k . We give an upper bound on the residue at s = 1 s=1 of ζ k \zeta _k which depends on β \beta , d k d_k and the behavior in k k of the rational primes p ∈ S p\in S . By combining these two results, we obtain lower bounds for the relative class numbers of some normal CM-fields K K which depend on the behavior in K K of the rational primes p ∈ S p\in S . 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.