Some explicit upper bounds for residues of zeta functions of number fields taking into account the behavior of the prime 2

2007 ◽  
Vol 125 (1) ◽  
pp. 43-67 ◽  
Author(s):  
Stéphane R. Louboutin
Keyword(s):  
Zeta Functions ◽  
Number Fields ◽  
Upper Bounds ◽  
Prime 2

Estimates of lattice points in the discriminant aspect over abelian extension fields

2018 ◽  
Vol 30 (3) ◽  
pp. 767-773 ◽  
Author(s):  
Wataru Takeda ◽  
Shin-ya Koyama
Keyword(s):  
Number Field ◽  
Zeta Functions ◽  
Number Fields ◽  
Upper Bounds ◽  
Abelian Extension ◽  
Lattice Points ◽  
Abelian Extensions ◽  
Lindelöf Hypothesis ◽  
Error Terms

AbstractWe estimate the number of relatively r-prime lattice points in {K^{m}} with their components having a norm less than x, where K is a number field. The error terms are estimated in terms of x and the discriminant D of the field K, as both x and D grow. The proof uses the bounds of Dedekind zeta functions. We obtain uniform upper bounds as K runs through number fields of any degree under assuming the Lindelöf hypothesis. We also show unconditional results for abelian extensions with a degree less than or equal to 6.

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

2001 ◽  
Vol 53 (6) ◽  
pp. 1194-1222 ◽  
Author(s):  
Stéphane Louboutin
Keyword(s):  
Lower Bounds ◽  
Class Number ◽  
Zeta Functions ◽  
Number Fields ◽  
Upper Bounds ◽  
Class Numbers ◽  
Class Groups ◽  
Relative Class ◽  
Relative Class Number ◽  
Root Discriminants

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.

Number fields with large minimal index containing quadratic subfields

2018 ◽  
Vol 14 (09) ◽  
pp. 2333-2342 ◽  
Author(s):  
Henry H. Kim ◽  
Zack Wolske
Keyword(s):  
Number Field ◽  
Number Fields ◽  
Upper Bounds ◽  
Minimal Index

In this paper, we consider number fields containing quadratic subfields with minimal index that is large relative to the discriminant of the number field. We give new upper bounds on the minimal index, and construct families with the largest possible minimal index.

Prime splitting in abelian number fields and linear combinations of Dirichlet characters

2014 ◽  
Vol 10 (04) ◽  
pp. 885-903 ◽  
Author(s):  
Paul Pollack
Keyword(s):  
Number Field ◽  
Nonzero Element ◽  
Number Fields ◽  
Upper Bounds ◽  
Prime Ideals ◽  
Linear Combinations ◽  
Dirichlet Characters ◽  
Abelian Number Field ◽  
Splitting Type ◽  
A Finite Group

Let 𝕏 be a finite group of primitive Dirichlet characters. Let ξ = ∑χ∈𝕏 aχ χ be a nonzero element of the group ring ℤ[𝕏]. We investigate the smallest prime q that is coprime to the conductor of each χ ∈ 𝕏 and that satisfies ∑χ∈𝕏 aχ χ(q) ≠ 0. Our main result is a nontrivial upper bound on q valid for certain special forms ξ. From this, we deduce upper bounds on the smallest unramified prime with a given splitting type in an abelian number field. For example, let K/ℚ be an abelian number field of degree n and conductor f. Let g be a proper divisor of n. If there is any unramified rational prime q that splits into g distinct prime ideals in ØK, then the least such q satisfies [Formula: see text].

scholarly journals The least unramified prime which does not split completely

2018 ◽  
Vol 30 (3) ◽  
pp. 651-661 ◽  
Author(s):  
Asif Zaman
Keyword(s):  
Prime Ideal ◽  
Finite Extension ◽  
Number Fields ◽  
Upper Bounds ◽  
Effective Field

AbstractLet {K/F} be a finite extension of number fields of degree {n\geq 2}. We establish effective field-uniform unconditional upper bounds for the least norm of a prime ideal {\mathfrak{p}} of F which is degree 1 over {\mathbb{Q}} and does not ramify or split completely in K. We improve upon the previous best known general estimates due to Li [7] when {F=\mathbb{Q}}, and Murty and Patankar [9] when {K/F} is Galois. Our bounds are the first when {K/F} is not assumed to be Galois and {F\neq\mathbb{Q}}.

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