Search

We study oscillations of the remainder term corresponding to the counting functions of the sets of elements with unique factorization length in semigroups of algebraic numbers such as the semigroup of algebraic integers or totally positive algebraic integers in a given normal field K. The results imply existence of oscillations when the exponent of the class group of the semigroup in question is sufficiently large depending on the field's degree. In particular, when K is a quadratic field or a normal cubic field oscillations exist whenever the class group is not isomorphic to C 2 a ⊕ C 3 b ⊕ C 4 c for nonnegative integers a,b,c. The main part of this study is concerned with the problem of multiplicative independence of Hecke zeta functions. We also show that there are infinitely many fields whose Dedekind zeta function has infinitely many nontrivial multiple zeros.

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.

In the adelic definition of the zeta function by Tate and Iwasawa, especially in the form given by Weil, one uses all Schwartz-Bruhat functions as "test functions"; we have found that such an adelic zeta function relative to Q contains the Dedekind zeta function of any finite normal extension of Q and that the normality assumption can be removed if Artin's conjecture is true. Introduction. We shall first review the definition of the zeta distribution associated with a number field K: let AK resp. A' denote the adele resp. idele groups of K, d Xx a Haar measure on A', and S (AK) the Schwartz-Bruhat space of the locally compact additive group AK; the topological dual S (AK)' of S (AK) is then the space of tempered distributions on AK. Let {xl denote the modulus of an idele x; then