A note on Dedekind zeta values at 1/2

2021 ◽  
pp. 1-11
Author(s):  
Neelam Kandhil
Keyword(s):  
Number Theory ◽  
Zeta Function ◽  
Number Field ◽  
Dedekind Zeta Function ◽  
Derivative Formula ◽  
Zeta Values

For a number field [Formula: see text], let [Formula: see text] be the Dedekind zeta function associated to [Formula: see text]. In this paper, we study non-vanishing and transcendence of [Formula: see text] as well as its derivative [Formula: see text] at [Formula: see text]. En route, we strengthen a result proved by Ram Murty and Tanabe [On the nature of [Formula: see text] and non-vanishing of [Formula: see text]-series at [Formula: see text], J. Number Theory 161 (2016) 444–456].

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 EXPLICIT ZERO-FREE REGIONS FOR DEDEKIND ZETA FUNCTIONS

2012 ◽  
Vol 08 (01) ◽  
pp. 125-147 ◽  
Author(s):  
HABIBA KADIRI
Keyword(s):  
Zeta Function ◽  
Number Field ◽  
Zeta Functions ◽  
Absolute Value ◽  
Dedekind Zeta Function ◽  
The Absolute

Let K be a number field, nK be its degree, and dK be the absolute value of its discriminant. We prove that, if dK is sufficiently large, then the Dedekind zeta function ζK(s) has no zeros in the region: [Formula: see text], [Formula: see text], where log M = 12.55 log dK + 9.69nK log |ℑ𝔪 s| + 3.03 nK + 58.63. Moreover, it has at most one zero in the region:[Formula: see text], [Formula: see text]. This zero if it exists is simple and is real. This argument also improves a result of Stark by a factor of 2: ζK(s) has at most one zero in the region [Formula: see text], [Formula: see text].

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 UNE MINORATION POUR L'EXPOSANT DU GROUPE DES CLASSES D'UN CORPS ENGENDRÉ PAR UN NOMBRE DE SALEM

2007 ◽  
Vol 03 (02) ◽  
pp. 217-229 ◽  
Author(s):  
FRANCESCO AMOROSO
Keyword(s):  
Zeta Function ◽  
Number Field ◽  
Lower Bounds ◽  
Riemann Hypothesis ◽  
Class Group ◽  
Relative Size ◽  
Generalized Riemann Hypothesis ◽  
Dedekind Zeta Function ◽  
Salem Number

In this article we extend the main result of [2] concerning lower bounds for the exponent of the class group of CM-fields. We consider a number field K generated by a Salem number α. If k denotes the field fixed by α ↦ α-1 we prove, under the generalized Riemann hypothesis for the Dedekind zeta function of K, lower bounds for the relative exponent eK/k and the relative size hK/k of the class group of K with respect to the class group of k.

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.

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 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.

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