ON SIMPLE ZEROS OF THE DEDEKIND ZETA‐FUNCTION OF A QUADRATIC NUMBER FIELD

Mathematika ◽  
2019 ◽  
Vol 65 (4) ◽  
pp. 851-861
Author(s):  
Xiaosheng Wu ◽  
Lilu Zhao
Keyword(s):  
Zeta Function ◽  
Number Field ◽  
Quadratic Number ◽  
Quadratic Number Field ◽  
Simple Zeros ◽  
Dedekind Zeta Function

Simple zeros of the zeta function of a quadratic number field. I

1986 ◽  
Vol 86 (3) ◽  
pp. 563-576 ◽  
Author(s):  
J. B. Conrey ◽  
A. Ghosh ◽  
S. M. Gonek
Keyword(s):  
Zeta Function ◽  
Number Field ◽  
Quadratic Number ◽  
Quadratic Number Field ◽  
Simple Zeros

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 Higher genus theory

2020 ◽  
Author(s):  
Peter Koymans ◽  
Carlo Pagano
Keyword(s):  
Number Field ◽  
Quadratic Forms ◽  
Class Group ◽  
Quadratic Extension ◽  
Number Fields ◽  
Explicit Description ◽  
Quadratic Number ◽  
Quadratic Number Field ◽  
Genus Theory ◽  
Narrow Class

Abstract In $1801$, Gauss found an explicit description, in the language of binary quadratic forms, for the $2$-torsion of the narrow class group and dual narrow class group of a quadratic number field. This is now known as Gauss’s genus theory. In this paper, we extend Gauss’s work to the setting of multi-quadratic number fields. To this end, we introduce and parametrize the categories of expansion groups and expansion Lie algebras, giving an explicit description for the universal objects of these categories. This description is inspired by the ideas of Smith [ 16] in his recent breakthrough on Goldfeld’s conjecture and the Cohen–Lenstra conjectures. Our main result shows that the maximal unramified multi-quadratic extension $L$ of a multi-quadratic number field $K$ can be reconstructed from the set of generalized governing expansions supported in the set of primes that ramify in $K$. This provides a recursive description for the group $\textrm{Gal}(L/\mathbb{Q})$ and a systematic procedure to construct the field $L$. A special case of our main result gives an upper bound for the size of $\textrm{Cl}^{+}(K)[2]$.

Colourings of quasicrystals

1994 ◽  
Vol 72 (7-8) ◽  
pp. 442-452 ◽  
Author(s):  
R. V. Moody ◽  
J. Patera
Keyword(s):  
Number Field ◽  
Penrose Tiling ◽  
Quadratic Number ◽  
Quadratic Number Field ◽  
Arithmetic Properties ◽  
Weight Lattice

We introduce a notion of colouring the points of a quasicrystal analogous to the idea of colouring or grading of the points of a lattice. Our results apply to quasicrystals that can be coordinatized by the ring R of integers of the quadratic number field [Formula: see text] and provide a useful and wide ranging tool for determining of sub-quasicrystals of quasicrystals. Using the arithmetic properties of R we determine all possible finite colourings. As examples we discuss the 4-colours of vertices of a Penrose tiling arising as a subset of 5-colouring of an R lattice, and the 4-colouring of quasicrystals arising from the D6 weight lattice.

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].

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