Dedekind Zeta Function Research Articles

Explicit Upper Bounds for Residues of Dedekind Zeta Functions and Values ofL-Functions ats= 1, and Explicit Lower Bounds for Relative Class Numbers of CM-Fields

AbstractWe provide the reader with a uniform approach for obtaining various useful explicit upper bounds on residues of Dedekind zeta functions of numbers fields and on absolute values of values at$s=1$of$L$-series associated with primitive characters on ray class groups of number fields. To make it quite clear to the reader how useful such bounds are when dealing with class number problems for CM-fields, we deduce an upper bound for the root discriminants of the normal CM-fields with (relative) class number one.

A note on the mean value of the zeta and $L$-functions. X

The present note reports on an explicit spectral formula for the fourth moment of the Dedekind zeta function $\zeta_{\mathrm{F}}$ of the Gaussian number field $\mathrm{F} = \mathbf{Q}(i)$, and on a new version of the sum formula of Kuznetsov type for $\mathrm{PSL}_2(\mathbf{Z}[i])\backslash \mathrm{PSL}_2(\mathbf{C})$. Our explicit formula (Theorem 5, below) for $\zeta_{\mathrm{F}}$ gives rise to a solution to a problem that has been posed on p. 183 of [M3] and, more explicitly, in [M4]. Also, our sum formula (Theorem 4, below) is an answer to a problem raised in [M4] concerning the inversion of a spectral sum formula over the Picard group $\mathrm{PSL}_2(\mathbf{Z}[i])$ acting on the three dimensional hyperbolic space (the $K$-trivial situation). To solve this problem, it was necessary to include the $K$-nontrivial situation into consideration, which is analogous to what has been experienced in the modular case.

Localization of the first zero of the Dedekind zeta function

Using Weil’s explicit formula, we propose a method to compute low zeros of the Dedekind zeta function. As an application of this method, we compute the first zero of the Dedekind zeta function associated to totally complex fields of degree less than or equal to 30 having the smallest known discriminant.

Interval Estimation of Values of Some Zeta and L-Functions and Their Applications

Some aspects of interval estimation of the Riemann zeta-function, Dirichlet i>L-functions, and Dedekind zeta-functions are briefly described and their applications are considered.

Fourth power moment of Dedekind zeta-functions of real quadratic number fields with class number one

Fourth power moment of Dedekind zeta-functions of real quadratic number fields with class number one

Explicit Bounds for Residues of Dedekind Zeta Functions, Values of L-Functions at s=1, and Relative Class Numbers

We give explicit upper bounds for residues at s=1 of Dedekind zeta functions of number fields, for |L(1, χ)| for nontrivial primitive characters χ on ray class groups, and for relative class numbers of CM fields. We also give explicit lower bounds for relative class numbers of CM fields (which do not contain any imaginary quadratic subfield). Most of these bounds stem from a useful lemma which enables us to easily obtain neat bounds.

Analytic Properties of Zeta Functions and Subgroup Growth

It has become somewhat of a cottage industry over the last fifteen years to understand the rate of growth of the number of subgroups of finite index in a group G. Although the story began much before, the recent activity grew out of a paper by Dan Segal in [36]. The story so far has been well-documented in Lubotzky’s subsequent survey paper in [30]. In [24] the second author of this article, Segal and Smith introduced the zeta function of a group as a tool for understanding this growth of subgroups. Let an(G) be the number of subgroups of index n in the finitely generated group G and sN (G) = a1(G) + · · ·+ aN (G) be the number of subgroups of index N or less. The zeta function is defined as the Dirichlet series with coefficients an(G) and has a natural interpretation as a noncommutative generalization of the Dedekind zeta function of a number field:

Values of zeta functions and class number 1 criterion for the simplest cubic fields

AbstractLet K be the simplest cubic field defined by the irreducible polynomial where m is a nonnegative rational integer such that m2 + 3m + 9 is square-free. We estimate the value of the Dedekind zeta function ζK(s) at s = −1 and get class number 1 criterion for the simplest cubic fields.

Determinantal Formula for the Special Values of the Dedekind Zeta Function of the Cyclotomic Field

A family of integer matrices, which generalize the Demjanenko matrix and the matrix defined by Silverman, is introduced and shown to compute the values at negative integers of the Dedekind zeta function of thepth cyclotomic field for any primep.

Tables of octic fields with a quartic subfield

We describe the computation of extended tables of degree 8 fields with a quartic subfield, using class field theory. In particular we find the minimum discriminants for all signatures and for all the possible Galois groups. We also discuss some phenomena and statistics discovered while making the tables, such as the occurrence of 11 non-isomorphic number fields having the same discriminant, or several pairs of non-isomorphic number fields having the same Dedekind zeta function.

Fourier coefficients of Eisenstein series of the exceptional group of type G2

K(s) F (s) of the two Dedekind zeta functions is an entire function in the complex variable s. From the point of view of the trace formula, the above basic question is expected to be equivalent to a basic question in automorphic L-functions, which asks whether or not the ratio L S ( _ ;s) S F (s) is entire for all irreducible cuspidal automorphic representation of GL(n;AF ) with trivial central character, where L S ( _ ;s ) is the standard tensor product L-function of with its contragredient _ , see for example the work of Jacquet and Zagier [JaZa]. The main idea in this paper is to develop two intrinsically related methods to attack the above two questions. The work of Siegel [Sie], and of Shimura [Shi] (and of Gelbart and Jacquet [GeJa]) provided an evidence for this approach for the case of n =2 . Combined with the work of Ginzburg [Gin], the main result of this paper shows that our approach works for the case of n =3 . It is hoped that such an approach extends to at least the case of n =5 .

The Positivity of a Sequence of Numbers and the Riemann Hypothesis

In this note, we prove that the Riemann hypothesis for the Dedekind zeta function is equivalent to the nonnegativity of a sequence of real numbers.

Zeros of Dedekind zeta functions in the critical strip

In this paper, we describe a computation which established the GRH to height 92 92 (resp. 40 40 ) for cubic number fields (resp. quartic number fields) with small discriminant. We use a method due to E. Friedman for computing values of Dedekind zeta functions, we take care of accumulated roundoff error to obtain results which are mathematically rigorous, and we generalize Turing’s criterion to prove that there is no zero off the critical line. We finally give results concerning the GRH for cubic and quartic fields, tables of low zeros for number fields of degree 5 5 and 6 6 , and statistics about the smallest zero of a number field.

Determination of all Quaternion Octic CM-Fields with Class Number 2

It is known that there are only finitely many normal CM-fields with class number one or with given class number (see [9, Theorem 2; 11, Theorem 2]) and J. Hoffstein showed that the degree of any normal CMfield with class number one is less than 436 (see [2, Corollary 2]). Moreover, K. Yamamura has determined all the abelian CM-fields with class number one: there are 172 non-isomorphic such number fields. In a recent paper the author and R. Okazaki moved on to the determination of non-abelian but normal octic CM-fields with class number one. Noticing that their class numbers are always even, they got rid of quaternion octic CM-fields, then they focussed on dihedral octic CM-fields and proved that there are 17 dihedral octic CM-fields with class number one. The aim of this paper is to get back to the quaternion case: we shall show that there exists exactly one quaternion octic CM-field with class number 2, namely: Q(V<x) wi tha = ( 2 + y'2)(3 + V3). Moreover, we shall show that the Hilbert class field of this number field is a normal and non-abelian CM-field of degree 16 with class number one. 1. Factorization of the Dedekind zeta function of a quaternion octic number field Let N be a quaternion octic CM-field, that is, N/Q is a normal octic extension with Galois group Gal (N/Q) the quaternion group Q8 with eight elements Q8 = {± 1, ± U ±j, ± k} with i = / = k = — 1 and ij = —ji = kjk = — kj = i and hi = — ik = j . Let N be the quartic subfield of N, so that N is the maximal totally real subfield of N. Let ki5 k and kfc be the three quadratic subfields of N, so that these quadratic fields are real quadratic fields. The quartic extensions N/k4, N/k; and N/kfc are cyclic and the quartic extension N/Q is biquadratic bicyclic. The lattice of subfields can be set out as follows:

Non-Galois cubic fields which are Euclidean but not norm-Euclidean

Weinberger in 1973 has shown that under the Generalized Riemann Hypothesis for Dedekind zeta functions, an algebraic number field with infinite unit group is Euclidean if and only if it is a principal ideal domain. Using a method recently introduced by us, we give two examples of cubic fields which are Euclidean but not norm-Euclidean.

A Note on Bernoulli Numbers and Shintani Generalized Bernoulli Polynomials

Generalized Bernoulli polynomials were introduced by Shintani in 1976 in order to express the special values at non-positive integers of Dedekind zeta functions for totally real numbers. The coefficients of such polynomials are finite combinations of products of Bernoulli numbers which are difficult to get hold of. On the other hand, Zagier was able to get the explicit formula for the special values in cases of real quadratic number fields. In this paper, we shall improve Shintani’s formula by proving that the special values can be determined by a finite set of polynomials. This provides a convenient way to evaluate the special values of various types of Dedekind functions. Indeed, a much broader class of zeta functions considered by the author [Minking Eie, The special values at negative integers of Dirichlet series associated with polynomials of several variables, Proceedings of A. M. S. 119 (1993), 51–61] admits a similar formula for its special values. As a consequence, we are able to find infinitely many identities among Bernoulli numbers through identities among zeta functions. All these identities are difficult to prove otherwise.

New Characterizations of Kronecker Equivalence

Two number fields K| k, K′| k are called Kronecker equivalent over k iff the sets of primes of k having a prime divisor of first degree in K respectively in K′ coincide up to at most finitely many exceptions. We give some new characterizations of Kronecker equivalence, which rely on a refinement (Theorem 3) of Klingen′s explicit calculation of M U ( F N| k ( P )) and on a result of Artin on representations of finite groups. Further we show how the Dirichlet series expansions of the Dedekind zeta functions of Kronecker equivalent fields are related. Moreover, our characterization gives information on the kernel of Artin L-functions and on class groups of Kronecker equivalent fields. Using character relations we define Kronecker equivalence and arithmetical equivalence for tupels of fields. This leads to a generalization of Artin′s theorem that over Q relations for zeta functions and character relations are the same to arbitrary ground fields; one only has to replace relations for zeta functions by some combinatorical conditions on residue degrees (Theorem 17).

Lower Bounds for Relative Class Numbers of CM-Fields

Let ${\mathbf {K}}$ be a CM-field that is a quadratic extension of a totally real number field ${\mathbf {k}}$. Under a technical assumption, we show that the relative class number of ${\mathbf {K}}$ is large compared with the absolute value of the discriminant of ${\mathbf {K}}$, provided that the Dedekind zeta function of ${\mathbf {k}}$ has a real zero $s$ such that $0 < s < 1$. This result will enable us to get sharp upper bounds on conductors of totally imaginary abelian number fields with class number one or with prescribed ideal class groups.

Lower bounds for relative class numbers of CM-fields

Let K {\mathbf {K}} be a CM-field that is a quadratic extension of a totally real number field k {\mathbf {k}} . Under a technical assumption, we show that the relative class number of K {\mathbf {K}} is large compared with the absolute value of the discriminant of K {\mathbf {K}} , provided that the Dedekind zeta function of k {\mathbf {k}} has a real zero s s such that 0 &gt; s &gt; 1 0 &gt; s &gt; 1 . This result will enable us to get sharp upper bounds on conductors of totally imaginary abelian number fields with class number one or with prescribed ideal class groups.

On a problem of R. Brauer for quotients of Dedekind zeta-functions

On a problem of R. Brauer for quotients of Dedekind zeta-functions