scholarly journals Extreme residues of Dedekind zeta functions

2017 ◽  
Vol 163 (2) ◽  
pp. 369-380 ◽  
Author(s):  
PETER J. CHO ◽  
HENRY H. KIM
Keyword(s):  
Lower Bounds ◽  
Zeta Functions ◽  
Infinite Family ◽  
Number Fields ◽  
Upper And Lower Bounds ◽  
Zero Set ◽  
Density Zero

AbstractIn a family ofSd+1-fields (d= 2, 3, 4), we obtain the conjectured upper and lower bounds of the residues of Dedekind zeta functions except for a density zero set. ForS5-fields, we need to assume the strong Artin conjecture. We also show that there exists an infinite family of number fields with the upper and lower bounds, resp.

Real zeros of Dedekind zeta functions

2015 ◽  
Vol 11 (03) ◽  
pp. 843-848 ◽  
Author(s):  
Stéphane R. Louboutin
Keyword(s):  
Real Axis ◽  
Lower Bounds ◽  
Zeta Functions ◽  
Number Fields ◽  
Class Numbers ◽  
The Real ◽  
Real Zeros ◽  
Relative Class ◽  
Free Region

Building on Stechkin and Kadiri's ideas we derive an explicit zero-free region of the real axis for Dedekind zeta functions of number fields. We then explain how this new region enables us to improve upon the previously known explicit lower bounds for class numbers of number fields and relative class numbers of CM-fields.

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.

scholarly journals Logarithmic derivatives of Artin -functions

2013 ◽  
Vol 149 (4) ◽  
pp. 568-586 ◽  
Author(s):  
Peter J. Cho ◽  
Henry H. Kim
Keyword(s):  
Galois Group ◽  
Number Field ◽  
Lower Bounds ◽  
Riemann Hypothesis ◽  
Number Fields ◽  
Upper And Lower Bounds ◽  
Zero Density ◽  
Galois Closures ◽  
Derivatives Of ◽  
Logarithmic Derivatives

AbstractLet $K$ be a number field of degree $n$, and let $d_K$ be its discriminant. Then, under the Artin conjecture, the generalized Riemann hypothesis and a certain zero-density hypothesis, we show that the upper and lower bounds of the logarithmic derivatives of Artin $L$-functions attached to $K$ at $s=1$ are $\log \log |d_K|$ and $-(n-1) \log \log |d_K|$, respectively. Unconditionally, we show that there are infinitely many number fields with the extreme logarithmic derivatives; they are families of number fields whose Galois closures have the Galois group $C_n$ for $n=2,3,4,6$, $D_n$ for $n=3,4,5$, $S_4$ or $A_5$.

ON THE GALOIS STRUCTURE OF ARITHMETIC COHOMOLOGY I: COMPACTLY SUPPORTED -ADIC COHOMOLOGY

2018 ◽  
Vol 239 ◽  
pp. 294-321
Author(s):  
DAVID BURNS
Keyword(s):  
Lower Bounds ◽  
Projective Module ◽  
Number Fields ◽  
Upper And Lower Bounds ◽  
Selmer Groups ◽  
Cohomology Groups ◽  
Field Extensions ◽  
Direct Sum ◽  
Galois Modules ◽  
Galois Structure

We investigate the Galois structures of $p$-adic cohomology groups of general $p$-adic representations over finite extensions of number fields. We show, in particular, that as the field extensions vary over natural families the Galois modules formed by these cohomology groups always decompose as the direct sum of a projective module and a complementary module of bounded $p$-rank. We use this result to derive new (upper and lower) bounds on the changes in ranks of Selmer groups over extensions of number fields and descriptions of the explicit Galois structures of natural arithmetic modules.

TORSION POINTS ON ELLIPTIC CURVES WITH COMPLEX MULTIPLICATION (WITH AN APPENDIX BY ALEX RICE)

2012 ◽  
Vol 09 (02) ◽  
pp. 447-479 ◽  
Author(s):  
PETE L. CLARK ◽  
BRIAN COOK ◽  
JAMES STANKEWICZ
Keyword(s):  
Elliptic Curves ◽  
Lower Bounds ◽  
Prime Order ◽  
Complex Multiplication ◽  
Number Fields ◽  
Upper And Lower Bounds ◽  
Torsion Points ◽  
Asymptotically Optimal ◽  
Large N ◽  
Cm Points

We present seven theorems on the structure of prime order torsion points on CM elliptic curves defined over number fields. The first three results refine bounds of Silverberg and Prasad–Yogananda by taking into account the class number of the CM order and the splitting of the prime in the CM field. In many cases we can show that our refined bounds are optimal or asymptotically optimal. We also derive asymptotic upper and lower bounds on the least degree of a CM-point on X1(N). Upon comparison to bounds for the least degree for which there exist infinitely many rational points on X1(N), we deduce that, for sufficiently large N, X1(N) will have a rational CM point of degree smaller than the degrees of at least all but finitely many non-CM points.

scholarly journals On the Generalized Distance Energy of Graphs

Mathematics ◽  
2019 ◽  
Vol 8 (1) ◽  
pp. 17 ◽  
Author(s):  
Abdollah Alhevaz ◽  
Maryam Baghipur ◽  
Hilal A. Ganie ◽  
Yilun Shang
Keyword(s):  
Lower Bounds ◽  
Complete Graph ◽  
Connected Graph ◽  
Distance Matrix ◽  
Wiener Index ◽  
Diagonal Matrix ◽  
Upper And Lower Bounds ◽  
Star Graph ◽  
Extremal Graphs ◽  
Generalized Distance

The generalized distance matrix D α ( G ) of a connected graph G is defined as D α ( G ) = α T r ( G ) + ( 1 − α ) D ( G ) , where 0 ≤ α ≤ 1 , D ( G ) is the distance matrix and T r ( G ) is the diagonal matrix of the node transmissions. In this paper, we extend the concept of energy to the generalized distance matrix and define the generalized distance energy E D α ( G ) . Some new upper and lower bounds for the generalized distance energy E D α ( G ) of G are established based on parameters including the Wiener index W ( G ) and the transmission degrees. Extremal graphs attaining these bounds are identified. It is found that the complete graph has the minimum generalized distance energy among all connected graphs, while the minimum is attained by the star graph among trees of order n.

scholarly journals Hadamard k-fractional inequalities of Fejér type for GA-s-convex mappings and applications

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Hui Lei ◽  
Gou Hu ◽  
Zhi-Jie Cao ◽  
Ting-Song Du
Keyword(s):  
Lower Bounds ◽  
Hypergeometric Functions ◽  
Upper And Lower Bounds ◽  
Weighted Inequalities ◽  
Convex Mappings ◽  
Special Means

Abstract The main aim of this paper is to establish some Fejér-type inequalities involving hypergeometric functions in terms of GA-s-convexity. For this purpose, we construct a Hadamard k-fractional identity related to geometrically symmetric mappings. Moreover, we give the upper and lower bounds for the weighted inequalities via products of two different mappings. Some applications of the presented results to special means are also provided.

scholarly journals On the Second-Largest Reciprocal Distance Signless Laplacian Eigenvalue

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 512
Author(s):  
Maryam Baghipur ◽  
Modjtaba Ghorbani ◽  
Hilal A. Ganie ◽  
Yilun Shang
Keyword(s):  
Lower Bounds ◽  
Complete Graph ◽  
Distance Matrix ◽  
Upper And Lower Bounds ◽  
Laplacian Eigenvalue ◽  
Signless Laplacian ◽  
Graph Parameters ◽  
Simple Connected Graph ◽  
Second Largest Eigenvalue ◽  
Reciprocal Distance Matrix

The signless Laplacian reciprocal distance matrix for a simple connected graph G is defined as RQ(G)=diag(RH(G))+RD(G). Here, RD(G) is the Harary matrix (also called reciprocal distance matrix) while diag(RH(G)) represents the diagonal matrix of the total reciprocal distance vertices. In the present work, some upper and lower bounds for the second-largest eigenvalue of the signless Laplacian reciprocal distance matrix of graphs in terms of various graph parameters are investigated. Besides, all graphs attaining these new bounds are characterized. Additionally, it is inferred that among all connected graphs with n vertices, the complete graph Kn and the graph Kn−e obtained from Kn by deleting an edge e have the maximum second-largest signless Laplacian reciprocal distance eigenvalue.

scholarly journals Multilevel Monte Carlo methods and lower–upper bounds in initial margin computations

2020 ◽  
Vol 26 (2) ◽  
pp. 131-161
Author(s):  
Florian Bourgey ◽  
Stefano De Marco ◽  
Emmanuel Gobet ◽  
Alexandre Zhou
Keyword(s):  
Monte Carlo ◽  
Lower Bounds ◽  
Asymptotic Optimality ◽  
Upper Bounds ◽  
Upper And Lower Bounds ◽  
Independent Random Variables ◽  
Outer Function ◽  
Control Problems ◽  
Multilevel Monte Carlo ◽  
Primal Dual

AbstractThe multilevel Monte Carlo (MLMC) method developed by M. B. Giles [Multilevel Monte Carlo path simulation, Oper. Res. 56 2008, 3, 607–617] has a natural application to the evaluation of nested expectations {\mathbb{E}[g(\mathbb{E}[f(X,Y)|X])]}, where {f,g} are functions and {(X,Y)} a couple of independent random variables. Apart from the pricing of American-type derivatives, such computations arise in a large variety of risk valuations (VaR or CVaR of a portfolio, CVA), and in the assessment of margin costs for centrally cleared portfolios. In this work, we focus on the computation of initial margin. We analyze the properties of corresponding MLMC estimators, for which we provide results of asymptotic optimality; at the technical level, we have to deal with limited regularity of the outer function g (which might fail to be everywhere differentiable). Parallel to this, we investigate upper and lower bounds for nested expectations as above, in the spirit of primal-dual algorithms for stochastic control problems.

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