EXPLICIT LOWER BOUNDS FOR RESIDUES AT s = 1 OF DEDEKIND ZETA FUNCTIONS AND RELATIVE CLASS NUMBERS OF CM-FIELDS

Let S be a given set of positive rational primes. Assume that the value of the Dedekind zeta function ζK of a number field K is less than or equal to zero at some real point β in the range 1 2 < β < 1. We give explicit lower bounds on the residue at s = 1 of this Dedekind zeta function which depend on β, the absolute value dK of the discriminant of K and the behavior in K of the rational primes p ∈ S. Now, let k be a real abelian number field and let β be any real zero of the zeta function of k. We give an upper bound on the residue at s = 1 of ζk which depends on β, dk and the behavior in k of the rational primes p ∈ S. By combining these two results, we obtain lower bounds for the relative class numbers of some normal CM-fields K which depend on the behavior in K of the rational primes p ∈ 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. The main results arrived at in this paper are Theorems 1, 14, 22 and 26. 1. Lower bounds for residues of zeta functions Let c > 0 be given (to be selected below). It has long been known that Hecke’s integral representations of Dedekind zeta functions ζK of number fields K can be used to obtain lower bounds for their residues κK at s = 1 of the type 1− (c/ log dK) ≤ β < 1 and ζK(β) ≤ 0 imply κK ≥ (1− β)d K (1 + o(1)), where o(1) is an error term that approaches zero as dK → ∞ provided that K ranges over number fields of a given degree (e.g. see [Lou2, Proposition A]. See also [Lan, Chapter XVI, Section 2, Lemma 3, p. 323] for a weaker result). Notice that the best lower bound one can deduce (for β = 1− (2/ log dK)) is of the type ζK(1− (2/ log dK)) ≤ 0 implies κK ≥ 2 e log dK (1 + o(1)). The first aim of this paper is to prove Theorem 1 below, which not only provides a nice treatment of this error term (by simply getting rid of it!) but also allows us to obtain lower bounds for these residues which depend on the behavior in K of a Received by the editors April 23, 2002 and, in revised form, January 6, 2003. 2000 Mathematics Subject Classification. Primary 11R42; Secondary 11R29.