A note on values of the Dedekind zeta-function at odd positive integers

2021 ◽  
pp. 1-12
Author(s):  
M. Ram Murty ◽  
Siddhi S. Pathak
Keyword(s):  
Zeta Function ◽  
Number Field ◽  
Algebraic Number Field ◽  
Number Fields ◽  
Imaginary Quadratic Fields ◽  
Fixed Integer ◽  
Dedekind Zeta Function ◽  
Totally Real ◽  
Positive Integers ◽  
Real Number Field

For an algebraic number field [Formula: see text], let [Formula: see text] be the associated Dedekind zeta-function. It is conjectured that [Formula: see text] is transcendental for any positive integer [Formula: see text]. The only known case of this conjecture was proved independently by Siegel and Klingen, namely that, when [Formula: see text] is a totally real number field, [Formula: see text] is an algebraic multiple of [Formula: see text] and hence, is transcendental. If [Formula: see text] is not totally real, the question of whether [Formula: see text] is irrational or not remains open. In this paper, we prove that for a fixed integer [Formula: see text], at most one of [Formula: see text] is rational, as [Formula: see text] varies over all imaginary quadratic fields. We also discuss a generalization of this theorem to CM-extensions of number fields.

scholarly journals ON THE MERTENS–CESÀRO THEOREM FOR NUMBER FIELDS

2015 ◽  
Vol 93 (2) ◽  
pp. 199-210 ◽  
Author(s):  
ANDREA FERRAGUTI ◽  
GIACOMO MICHELI
Keyword(s):  
Zeta Function ◽  
Number Field ◽  
Number Fields ◽  
Algebraic Integers ◽  
Ring Of Integers ◽  
Dedekind Zeta Function ◽  
Natural Density ◽  
Suitable Notion

Let $K$ be a number field with ring of integers ${\mathcal{O}}$. After introducing a suitable notion of density for subsets of ${\mathcal{O}}$, generalising the natural density for subsets of $\mathbb{Z}$, we show that the density of the set of coprime $m$-tuples of algebraic integers is $1/{\it\zeta}_{K}(m)$, where ${\it\zeta}_{K}$ is the Dedekind zeta function of $K$. This generalises a result found independently by Mertens [‘Ueber einige asymptotische Gesetze der Zahlentheorie’, J. reine angew. Math. 77 (1874), 289–338] and Cesàro [‘Question 75 (solution)’, Mathesis 3 (1883), 224–225] concerning the density of coprime pairs of integers in $\mathbb{Z}$.

scholarly journals Weak Arithmetic Equivalence

2015 ◽  
Vol 58 (1) ◽  
pp. 115-127 ◽  
Author(s):  
Guillermo Mantilla-Soler
Keyword(s):  
Zeta Function ◽  
Real Number ◽  
Number Field ◽  
Number Fields ◽  
Trace Form ◽  
Root Numbers ◽  
Local Root ◽  
Arithmetic Equivalence ◽  
Totally Real ◽  
Local Root Numbers

AbstractInspired by the invariant of a number field given by its zeta function, we define the notion of weak arithmetic equivalence and show that under certain ramification hypotheses this equivalence determines the local root numbers of the number field. This is analogous to a result of Rohrlich on the local root numbers of a rational elliptic curve. Additionally, we prove that for tame non-totally real number fields, the integral trace form is invariant under arithmetic equivalence

ON S-HARDY–LITTLEWOOD HOMOGENEOUS SPACES

1998 ◽  
Vol 09 (06) ◽  
pp. 723-757 ◽  
Author(s):  
MASANORI MORISHITA ◽  
TAKAO WATANABE
Keyword(s):  
Number Field ◽  
Homogeneous Spaces ◽  
Algebraic Number Field ◽  
Number Fields ◽  
Positive Definite ◽  
Integral Points ◽  
Distribution Property ◽  
Totally Positive ◽  
Finite Set ◽  
Totally Real

We study the asymptotic distribution of S-integral points on affine homogeneous spaces in the light of the Hardy–Littlewood property introduced by Borovoi and Rudnick. We introduce the S-Hardy–Littlewood property for affine homogeneous spaces defined over an algebraic number field and a finite set S of places of the base field. We work with the adelic harmonic analysis on affine algebraic groups over a number field to determine the asymptotic density of S-integral points under congruence conditions. We give some new examples of strongly or relatively S-Hardy–Littlewood homogeneous spaces over number fields. As an application, we prove certain asymptotically uniform distribution property of integral points on an ellipsoid defined by a totally positive definite tenary quadratic form over a totally real number field.

Hecke L-Functions and the Distribution of Totally Positive Integers

2007 ◽  
Vol 59 (4) ◽  
pp. 673-695 ◽  
Author(s):  
Avner Ash ◽  
Solomon Friedberg
Keyword(s):  
Real Number ◽  
Number Field ◽  
Expected Value ◽  
Fractional Ideal ◽  
Compact Torus ◽  
Positive Elements ◽  
Totally Positive ◽  
Totally Real ◽  
Positive Integers ◽  
Real Number Field

AbstractLet K be a totally real number field of degree n. We show that the number of totally positive integers (or more generally the number of totally positive elements of a given fractional ideal) of given trace is evenly distributed around its expected value, which is obtained fromgeometric considerations. This result depends on unfolding an integral over a compact torus.

Tame kernels and second regulators of number fields and their subfields

2013 ◽  
Vol 12 (1) ◽  
pp. 137-165 ◽  
Author(s):  
Jerzy Browkin ◽  
Herbert Gangl
Keyword(s):  
Zeta Function ◽  
Galois Group ◽  
Number Field ◽  
Dihedral Group ◽  
Number Fields ◽  
Numerical Results ◽  
Alternating Group ◽  
Dedekind Zeta Function ◽  
Tame Kernel ◽  
Tame Kernels

AbstractAssuming a version of the Lichtenbaum conjecture, we apply Brauer-Kuroda relations between the Dedekind zeta function of a number field and the zeta function of some of its subfields to prove formulas relating the order of the tame kernel of a number field F with the orders of the tame kernels of some of its subfields. The details are given for fields F which are Galois over ℚ with Galois group the group ℤ/2 × ℤ/2, the dihedral group D2p; p an odd prime, or the alternating group A4. We include numerical results illustrating these formulas.

scholarly journals Computation of L(0, χ) and of relative class numbers of CM-fields

2001 ◽  
Vol 161 ◽  
pp. 171-191 ◽  
Author(s):  
Stéphane Louboutin
Keyword(s):  
Real Number ◽  
Number Field ◽  
Class Group ◽  
Number Fields ◽  
Class Numbers ◽  
Finite Part ◽  
Relative Class ◽  
Relative Class Number ◽  
Totally Real ◽  
Real Number Field

Let χ be a nontrivial Hecke character on a (strict) ray class group of a totally real number field L of discriminant dL. Then, L(0, χ) is an algebraic number of some cyclotomic number field. We develop an efficient technique for computing the exact values at s = 0 of such abelian Hecke L-functions over totally real number fields L. Let fχ denote the norm of the finite part of the conductor of χ. Then, roughly speaking, we can compute L(0, χ) in O((dLfx)0.5+∊) elementary operations. We then explain how the computation of relative class numbers of CM-fields boils down to the computation of exact values at s = 0 of such abelian Hecke L-functions over totally real number fields L. Finally, we give examples of relative class number computations for CM-fields of large degrees based on computations of L(0, χ) over totally real number fields of degree 2 and 6.

ON THE EULER PRODUCT OF THE DEDEKIND ZETA FUNCTION

2009 ◽  
Vol 05 (02) ◽  
pp. 293-301
Author(s):  
XIAN-JIN LI
Keyword(s):  
Zeta Function ◽  
Algebraic Number ◽  
Zeta Functions ◽  
Algebraic Number Field ◽  
Number Fields ◽  
Product Formula ◽  
Euler Product ◽  
Dedekind Zeta Function ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

It is well known that the Euler product formula for the Riemann zeta function ζ(s) is still valid for ℜ(s) = 1 and s ≠ 1. In this paper, we extend this result to zeta functions of number fields. In particular, we show that the Dedekind zeta function ζk(s) for any algebraic number field k can be written as the Euler product on the line ℜ(s) = 1 except at the point s = 1. As a corollary, we obtain the Euler product formula on the line ℜ(s) = 1 for Dirichlet L-functions L(s, χ) of real characters.

scholarly journals Class-number problems for cubic number fields

1995 ◽  
Vol 138 ◽  
pp. 199-208 ◽  
Author(s):  
Stéphane Louboutin
Keyword(s):  
Zeta Function ◽  
Number Field ◽  
Class Number ◽  
Number Fields ◽  
Algebraic Integers ◽  
Absolute Value ◽  
Dedekind Zeta Function ◽  
Cubic Number Fields ◽  
Ring Of Algebraic Integers ◽  
The Absolute

Let M be any number field. We let DM, dM, hu, , AM and RegM be the discriminant, the absolute value of the discriminant, the class-number, the Dedekind zeta-function, the ring of algebraic integers and the regulator of M, respectively.we set If q is any odd prime we let (⋅/q) denote the Legendre’s symbol.

The (α, β)-ramification invariants of a number field

2021 ◽  
Vol 71 (1) ◽  
pp. 251-263
Author(s):  
Guillermo Mantilla-Soler
Keyword(s):  
Zeta Function ◽  
Number Field ◽  
Residue Class ◽  
Interesting Application ◽  
Dedekind Zeta Function ◽  
Arithmetic Properties

Abstract Let L be a number field. For a given prime p, we define integers α p L $ \alpha_{p}^{L} $ and β p L $ \beta_{p}^{L} $ with some interesting arithmetic properties. For instance, β p L $ \beta_{p}^{L} $ is equal to 1 whenever p does not ramify in L and α p L $ \alpha_{p}^{L} $ is divisible by p whenever p is wildly ramified in L. The aforementioned properties, although interesting, follow easily from definitions; however a more interesting application of these invariants is the fact that they completely characterize the Dedekind zeta function of L. Moreover, if the residue class mod p of α p L $ \alpha_{p}^{L} $ is not zero for all p then such residues determine the genus of the integral trace.

scholarly journals Hyperbolic tessellations and generators of for imaginary quadratic fields

2021 ◽  
Vol 9 ◽  
Author(s):  
David Burns ◽  
Rob de Jeu ◽  
Herbert Gangl ◽  
Alexander D. Rahm ◽  
Dan Yasaki
Keyword(s):  
Number Field ◽  
Number Fields ◽  
Quadratic Fields ◽  
Quadratic Number ◽  
Imaginary Quadratic Fields ◽  
Quadratic Number Fields ◽  
Formula Group ◽  
Imaginary Quadratic Number ◽  
Direct Calculations

Abstract We develop methods for constructing explicit generators, modulo torsion, of the $K_3$ -groups of imaginary quadratic number fields. These methods are based on either tessellations of hyperbolic $3$ -space or on direct calculations in suitable pre-Bloch groups and lead to the very first proven examples of explicit generators, modulo torsion, of any infinite $K_3$ -group of a number field. As part of this approach, we make several improvements to the theory of Bloch groups for $ K_3 $ of any field, predict the precise power of $2$ that should occur in the Lichtenbaum conjecture at $ -1 $ and prove that this prediction is valid for all abelian number fields.

Sign in / Sign up

Export Citation Format

Share Document