Multiple Zeta Functions Multiple Polylogarithms And Their Special Values

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Multiple Zeta Functions, Multiple Polylogarithms and Their Special Values

Jianqiang Zhao
Multiple Zeta Functions Multiple Polylogarithms and Their Special Values Book PDF

Author : Jianqiang Zhao
Publisher : World Scientific
Page : 620 pages
File Size : 41,8 Mb
Release : 2016-03-07
Category : Mathematics
ISBN : 9789814689410

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Multiple Zeta Functions, Multiple Polylogarithms and Their Special Values by Jianqiang Zhao Pdf

This is the first introductory book on multiple zeta functions and multiple polylogarithms which are the generalizations of the Riemann zeta function and the classical polylogarithms, respectively, to the multiple variable setting. It contains all the basic concepts and the important properties of these functions and their special values. This book is aimed at graduate students, mathematicians and physicists who are interested in this current active area of research. The book will provide a detailed and comprehensive introduction to these objects, their fascinating properties and interesting relations to other mathematical subjects, and various generalizations such as their q-analogs and their finite versions (by taking partial sums modulo suitable prime powers). Historical notes and exercises are provided at the end of each chapter. Contents:Multiple Zeta FunctionsMultiple Polylogarithms (MPLs)Multiple Zeta Values (MZVs)Drinfeld Associator and Single-Valued MZVsMultiple Zeta Value IdentitiesSymmetrized Multiple Zeta Values (SMZVs)Multiple Harmonic Sums (MHSs) and Alternating VersionFinite Multiple Zeta Values and Finite Euler Sumsq-Analogs of Multiple Harmonic (Star) Sums Readership: Advanced undergraduates and graduate students in mathematics, mathematicians interested in multiple zeta values. Key Features:For the first time, a detailed explanation of the theory of multiple zeta values is given in book form along with numerous illustrations in explicit examplesThe book provides for the first time a comprehensive introduction to multiple polylogarithms and their special values at roots of unity, from the basic definitions to the more advanced topics in current active researchThe book contains a few quite intriguing results relating the special values of multiple zeta functions and multiple polylogarithms to other branches of mathematics and physics, such as knot theory and the theory of motivesMany exercises contain supplementary materials which deepens the reader's understanding of the main text

Zeta Functions, Topology and Quantum Physics

Takashi Aoki,Shigeru Kanemitsu,Mikio Nakahara,Yasuo Ohno
Zeta Functions Topology and Quantum Physics Book PDF

Author : Takashi Aoki,Shigeru Kanemitsu,Mikio Nakahara,Yasuo Ohno
Publisher : Springer Science & Business Media
Page : 219 pages
File Size : 54,8 Mb
Release : 2008-05-10
Category : Mathematics
ISBN : 9780387249810

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Zeta Functions, Topology and Quantum Physics by Takashi Aoki,Shigeru Kanemitsu,Mikio Nakahara,Yasuo Ohno Pdf

This volume contains papers by invited speakers of the symposium "Zeta Functions, Topology and Quantum Physics" held at Kinki U- versity in Osaka, Japan, during the period of March 3-6, 2003. The aims of this symposium were to establish mutual understanding and to exchange ideas among researchers working in various fields which have relation to zeta functions and zeta values. We are very happy to add this volume to the series Developments in Mathematics from Springer. In this respect, Professor Krishnaswami Alladi helped us a lot by showing his keen and enthusiastic interest in publishing this volume and by contributing his paper with Alexander Berkovich. We gratefully acknowledge financial support from Kinki University. We would like to thank Professor Megumu Munakata, Vice-Rector of Kinki University, and Professor Nobuki Kawashima, Director of School of Interdisciplinary Studies of Science and Engineering, Kinki Univ- sity, for their interest and support. We also thank John Martindale of Springer for his excellent editorial work.

The Theory of Zeta-Functions of Root Systems

Yasushi Komori,Kohji Matsumoto,Hirofumi Tsumura
The Theory of Zeta Functions of Root Systems Book PDF

Author : Yasushi Komori,Kohji Matsumoto,Hirofumi Tsumura
Publisher : Springer Nature
Page : 419 pages
File Size : 49,8 Mb
Release : 2024-02-03
Category : Mathematics
ISBN : 9789819909100

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The Theory of Zeta-Functions of Root Systems by Yasushi Komori,Kohji Matsumoto,Hirofumi Tsumura Pdf

The contents of this book was created by the authors as a simultaneous generalization of Witten zeta-functions, Mordell–Tornheim multiple zeta-functions, and Euler–Zagier multiple zeta-functions. Zeta-functions of root systems are defined by certain multiple series, given in terms of root systems. Therefore, they intrinsically have the action of associated Weyl groups. The exposition begins with a brief introduction to the theory of Lie algebras and root systems and then provides the definition of zeta-functions of root systems, explicit examples associated with various simple Lie algebras, meromorphic continuation and recursive analytic structure described by Dynkin diagrams, special values at integer points, functional relations, and the background given by the action of Weyl groups. In particular, an explicit form of Witten’s volume formula is provided. It is shown that various relations among special values of Euler–Zagier multiple zeta-functions—which usually are called multiple zeta values (MZVs) and are quite important in connection with Zagier’s conjecture—are just special cases of various functional relations among zeta-functions of root systems. The authors further provide other applications to the theory of MZVs and also introduce generalizations with Dirichlet characters, and with certain congruence conditions. The book concludes with a brief description of other relevant topics.

The Theory of Multiple Zeta Values with Applications in Combinatorics

Minking Eie
The Theory of Multiple Zeta Values with Applications in Combinatorics Book PDF

Author : Minking Eie
Publisher : World Scientific
Page : 313 pages
File Size : 55,7 Mb
Release : 2013
Category : Mathematics
ISBN : 9789814472647

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The Theory of Multiple Zeta Values with Applications in Combinatorics by Minking Eie Pdf

This is the first book on the theory of multiple zeta values since its birth around 1994. Readers will find that the shuffle products of multiple zeta values are applied to complicated counting problems in combinatorics, producing numerous interesting identities that are ready to be used. This will provide a powerful tool to deal with problems in multiple zeta values, both in evaluations and shuffle relations. The volume will benefit graduate students doing research in number theory.

Zeta and Q-Zeta Functions and Associated Series and Integrals

H. M. Srivastava,Junesang Choi
Zeta and Q Zeta Functions and Associated Series and Integrals Book PDF

Author : H. M. Srivastava,Junesang Choi
Publisher : Elsevier
Page : 675 pages
File Size : 55,8 Mb
Release : 2011-10-25
Category : Mathematics
ISBN : 9780123852182

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Zeta and Q-Zeta Functions and Associated Series and Integrals by H. M. Srivastava,Junesang Choi Pdf

Zeta and q-Zeta Functions and Associated Series and Integrals is a thoroughly revised, enlarged and updated version of Series Associated with the Zeta and Related Functions. Many of the chapters and sections of the book have been significantly modified or rewritten, and a new chapter on the theory and applications of the basic (or q-) extensions of various special functions is included. This book will be invaluable because it covers not only detailed and systematic presentations of the theory and applications of the various methods and techniques used in dealing with many different classes of series and integrals associated with the Zeta and related functions, but stimulating historical accounts of a large number of problems and well-classified tables of series and integrals. Detailed and systematic presentations of the theory and applications of the various methods and techniques used in dealing with many different classes of series and integrals associated with the Zeta and related functions

Analytic Methods In Number Theory: When Complex Numbers Count

Wadim Zudilin
Analytic Methods In Number Theory When Complex Numbers Count Book PDF

Author : Wadim Zudilin
Publisher : World Scientific
Page : 192 pages
File Size : 42,8 Mb
Release : 2023-08-22
Category : Mathematics
ISBN : 9789811279331

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Analytic Methods In Number Theory: When Complex Numbers Count by Wadim Zudilin Pdf

There is no surprise that arithmetic properties of integral ('whole') numbers are controlled by analytic functions of complex variable. At the same time, the values of analytic functions themselves happen to be interesting numbers, for which we often seek explicit expressions in terms of other 'better known' numbers or try to prove that no such exist. This natural symbiosis of number theory and analysis is centuries old but keeps enjoying new results, ideas and methods.The present book takes a semi-systematic review of analytic achievements in number theory ranging from classical themes about primes, continued fractions, transcendence of π and resolution of Hilbert's seventh problem to some recent developments on the irrationality of the values of Riemann's zeta function, sizes of non-cyclotomic algebraic integers and applications of hypergeometric functions to integer congruences.Our principal goal is to present a variety of different analytic techniques that are used in number theory, at a reasonably accessible — almost popular — level, so that the materials from this book can suit for teaching a graduate course on the topic or for a self-study. Exercises included are of varying difficulty and of varying distribution within the book (some chapters get more than other); they not only help the reader to consolidate their understanding of the material but also suggest directions for further study and investigation. Furthermore, the end of each chapter features brief notes about relevant developments of the themes discussed.

Zeta Functions, Topology and Quantum Physics

Takashi Aoki,Shigeru Kanemitsu,Mikio Nakahara,Yasuo Ohno
Zeta Functions Topology and Quantum Physics Book PDF

Author : Takashi Aoki,Shigeru Kanemitsu,Mikio Nakahara,Yasuo Ohno
Publisher : Springer
Page : 0 pages
File Size : 53,5 Mb
Release : 2008-11-01
Category : Mathematics
ISBN : 0387522883

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Zeta Functions, Topology and Quantum Physics by Takashi Aoki,Shigeru Kanemitsu,Mikio Nakahara,Yasuo Ohno Pdf

This volume contains papers by invited speakers of the symposium "Zeta Functions, Topology and Quantum Physics" held at Kinki U- versity in Osaka, Japan, during the period of March 3-6, 2003. The aims of this symposium were to establish mutual understanding and to exchange ideas among researchers working in various fields which have relation to zeta functions and zeta values. We are very happy to add this volume to the series Developments in Mathematics from Springer. In this respect, Professor Krishnaswami Alladi helped us a lot by showing his keen and enthusiastic interest in publishing this volume and by contributing his paper with Alexander Berkovich. We gratefully acknowledge financial support from Kinki University. We would like to thank Professor Megumu Munakata, Vice-Rector of Kinki University, and Professor Nobuki Kawashima, Director of School of Interdisciplinary Studies of Science and Engineering, Kinki Univ- sity, for their interest and support. We also thank John Martindale of Springer for his excellent editorial work.

An Introduction to Non-Abelian Class Field Theory

Toyokazu Hiramatsu,Seiken Saito
An Introduction to Non Abelian Class Field Theory Book PDF

Author : Toyokazu Hiramatsu,Seiken Saito
Publisher : World Scientific
Page : 188 pages
File Size : 48,9 Mb
Release : 2016-09-13
Category : Mathematics
ISBN : 9789813142282

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An Introduction to Non-Abelian Class Field Theory by Toyokazu Hiramatsu,Seiken Saito Pdf

This monograph provides a brief exposition of automorphic forms of weight 1 and their applications to arithmetic, especially to Galois representations. One of the outstanding problems in arithmetic is a generalization of class field theory to non-abelian Galois extension of number fields. In this volume, we discuss some relations between this problem and cusp forms of weight 1.

Derived Langlands: Monomial Resolutions Of Admissible Representations

Victor P Snaith
Derived Langlands Monomial Resolutions Of Admissible Representations Book PDF

Author : Victor P Snaith
Publisher : World Scientific
Page : 356 pages
File Size : 45,6 Mb
Release : 2018-12-06
Category : Mathematics
ISBN : 9789813275768

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Derived Langlands: Monomial Resolutions Of Admissible Representations by Victor P Snaith Pdf

The Langlands Programme is one of the most important areas in modern pure mathematics. The importance of this volume lies in its potential to recast many aspects of the programme in an entirely new context. For example, the morphisms in the monomial category of a locally p-adic Lie group have a distributional description, due to Bruhat in his thesis. Admissible representations in the programme are often treated via convolution algebras of distributions and representations of Hecke algebras. The monomial embedding, introduced in this book, elegantly fits together these two uses of distribution theory. The author follows up this application by giving the monomial category treatment of the Bernstein Centre, classified by Deligne-Bernstein-Zelevinsky.This book gives a new categorical setting in which to approach well-known topics. Therefore, the context used to explain examples is often the more generally accessible case of representations of finite general linear groups. For example, Galois base-change and epsilon factors for locally p-adic Lie groups are illustrated by the analogous Shintani descent and Kondo-Gauss sums, respectively. General linear groups of local fields are emphasized. However, since the philosophy of this book is essentially that of homotopy theory and algebraic topology, it includes a short appendix showing how the buildings of Bruhat-Tits, sufficient for the general linear group, may be generalised to the tom Dieck spaces (now known as the Baum-Connes spaces) when G is a locally p-adic Lie group.The purpose of this monograph is to describe a functorial embedding of the category of admissible k-representations of a locally profinite topological group G into the derived category of the additive category of the admissible k-monomial module category. Experts in the Langlands Programme may be interested to learn that when G is a locally p-adic Lie group, the monomial category is closely related to the category of topological modules over a sort of enlarged Hecke algebra with generators corresponding to characters on compact open modulo the centre subgroups of G. Having set up this functorial embedding, how the ingredients of the celebrated Langlands Programme adapt to the context of the derived monomial module category is examined. These include automorphic representations, epsilon factors and L-functions, modular forms, Weil-Deligne representations, Galois base change and Hecke operators.

Problems and Solutions in Real Analysis

Masayoshi Hata
Problems and Solutions in Real Analysis Book PDF

Author : Masayoshi Hata
Publisher : World Scientific Publishing Company
Page : 376 pages
File Size : 42,7 Mb
Release : 2016-12-12
Category : Mathematics
ISBN : 9789813142848

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Problems and Solutions in Real Analysis by Masayoshi Hata Pdf

This second edition introduces an additional set of new mathematical problems with their detailed solutions in real analysis. It also provides numerous improved solutions to the existing problems from the previous edition, and includes very useful tips and skills for the readers to master successfully. There are three more chapters that expand further on the topics of Bernoulli numbers, differential equations and metric spaces. Each chapter has a summary of basic points, in which some fundamental definitions and results are prepared. This also contains many brief historical comments for some significant mathematical results in real analysis together with many references. Problems and Solutions in Real Analysis can be treated as a collection of advanced exercises by undergraduate students during or after their courses of calculus and linear algebra. It is also instructive for graduate students who are interested in analytic number theory. Readers will also be able to completely grasp a simple and elementary proof of the Prime Number Theorem through several exercises. This volume is also suitable for non-experts who wish to understand mathematical analysis. Request Inspection Copy Contents:Sequences and LimitsInfinite SeriesContinuous FunctionsDifferentiationIntegrationImproper IntegralsSeries of FunctionsApproximation by PolynomialsConvex FunctionsVarious Proof ζ(2) = π2/6Functions of Several VariablesUniform DistributionRademacher FunctionsLegendre PolynomialsChebyshev PolynomialsGamma FunctionPrime Number TheoremBernoulli NumbersMetric SpacesDifferential Equations Readership: Undergraduates and graduate students in mathematical analysis.

Elementary Modular Iwasawa Theory

Haruzo Hida
Elementary Modular Iwasawa Theory Book PDF

Author : Haruzo Hida
Publisher : World Scientific
Page : 446 pages
File Size : 43,6 Mb
Release : 2021-10-04
Category : Mathematics
ISBN : 9789811241383

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Elementary Modular Iwasawa Theory by Haruzo Hida Pdf

This book is the first to provide a comprehensive and elementary account of the new Iwasawa theory innovated via the deformation theory of modular forms and Galois representations. The deformation theory of modular forms is developed by generalizing the cohomological approach discovered in the author's 2019 AMS Leroy P Steele Prize-winning article without using much algebraic geometry.Starting with a description of Iwasawa's classical results on his proof of the main conjecture under the Kummer-Vandiver conjecture (which proves cyclicity of his Iwasawa module more than just proving his main conjecture), we describe a generalization of the method proving cyclicity to the adjoint Selmer group of every ordinary deformation of a two-dimensional Artin Galois representation.The fundamentals in the first five chapters are as follows:Many open problems are presented to stimulate young researchers pursuing their field of study.

Smooth-automorphic Forms And Smooth-automorphic Representations

Harald Grobner
Smooth automorphic Forms And Smooth automorphic Representations Book PDF

Author : Harald Grobner
Publisher : World Scientific
Page : 262 pages
File Size : 52,9 Mb
Release : 2023-06-09
Category : Mathematics
ISBN : 9789811246180

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Smooth-automorphic Forms And Smooth-automorphic Representations by Harald Grobner Pdf

This book provides a conceptual introduction into the representation theory of local and global groups, with final emphasis on automorphic representations of reductive groups G over number fields F.Our approach to automorphic representations differs from the usual literature: We do not consider 'K-finite' automorphic forms, but we allow a richer class of smooth functions of uniform moderate growth. Contrasting the usual approach, our space of 'smooth-automorphic forms' is intrinsic to the group scheme G/F.This setup also covers the advantage that a perfect representation-theoretical symmetry between the archimedean and non-archimedean places of the number field F is regained, by making the bigger space of smooth-automorphic forms into a proper, continuous representation of the full group of adelic points of G.Graduate students and researchers will find the covered topics appear for the first time in a book, where the theory of smooth-automorphic representations is robustly developed and presented in great detail.

Contemporary Trends In Algebraic Geometry And Algebraic Topology

Shiing-shen Chern,Richard Hain,Lei Fu
Contemporary Trends In Algebraic Geometry And Algebraic Topology Book PDF

Author : Shiing-shen Chern,Richard Hain,Lei Fu
Publisher : World Scientific
Page : 276 pages
File Size : 47,8 Mb
Release : 2002-08-02
Category : Mathematics
ISBN : 9789814488518

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Contemporary Trends In Algebraic Geometry And Algebraic Topology by Shiing-shen Chern,Richard Hain,Lei Fu Pdf

The Wei-Liang Chow and Kuo-Tsai Chen Memorial Conference was proposed and held by Prof S S Chern in Nankai Institute of Mathematics. It was devoted to memorializing those two outstanding and original Chinese mathematicians who had made significant contributions to algebraic geometry and algebraic topology, respectively. It also provided a forum for leading mathematicians to expound and discuss their views on new ideas in these fields, as well as trends in 21st Century mathematics. About 100 mathematicians participated in the conference, including Sir Michael Atiyah, Jacob Palis, Phillip Griffiths, David Eisenbud, Philippe Tondeur, Yujiro Kawamata, Tian Gang, etc.This invaluable volume contains the selected papers presented at the conference. The topics include canonical maps of Gorenstein 3-folds, fundamental groups of algebraic curves, Chen's interated integrals, algebraic fiber spaces, and others.

Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory

Johannes Blümlein,Carsten Schneider,Peter Paule
Elliptic Integrals Elliptic Functions and Modular Forms in Quantum Field Theory Book PDF

Author : Johannes Blümlein,Carsten Schneider,Peter Paule
Publisher : Springer
Page : 509 pages
File Size : 53,8 Mb
Release : 2019-01-30
Category : Computers
ISBN : 9783030044800

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Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory by Johannes Blümlein,Carsten Schneider,Peter Paule Pdf

This book includes review articles in the field of elliptic integrals, elliptic functions and modular forms intending to foster the discussion between theoretical physicists working on higher loop calculations and mathematicians working in the field of modular forms and functions and analytic solutions of higher order differential and difference equations.

$q$-Series with Applications to Combinatorics, Number Theory, and Physics

Bruce C. Berndt,Ken Ono
q Series with Applications to Combinatorics Number Theory and Physics Book PDF

Author : Bruce C. Berndt,Ken Ono
Publisher : American Mathematical Soc.
Page : 290 pages
File Size : 42,9 Mb
Release : 2001
Category : q-series
ISBN : 9780821827468

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$q$-Series with Applications to Combinatorics, Number Theory, and Physics by Bruce C. Berndt,Ken Ono Pdf

The subject of $q$-series can be said to begin with Euler and his pentagonal number theorem. In fact, $q$-series are sometimes called Eulerian series. Contributions were made by Gauss, Jacobi, and Cauchy, but the first attempt at a systematic development, especially from the point of view of studying series with the products in the summands, was made by E. Heine in 1847. In the latter part of the nineteenth and in the early part of the twentieth centuries, two Englishmathematicians, L. J. Rogers and F. H. Jackson, made fundamental contributions. In 1940, G. H. Hardy described what we now call Ramanujan's famous $ 1\psi 1$ summation theorem as ``a remarkable formula with many parameters.'' This is now one of the fundamental theorems of the subject. Despite humble beginnings,the subject of $q$-series has flourished in the past three decades, particularly with its applications to combinatorics, number theory, and physics. During the year 2000, the University of Illinois embraced The Millennial Year in Number Theory. One of the events that year was the conference $q$-Series with Applications to Combinatorics, Number Theory, and Physics. This event gathered mathematicians from the world over to lecture and discuss their research. This volume presents nineteen of thepapers presented at the conference. The excellent lectures that are included chart pathways into the future and survey the numerous applications of $q$-series to combinatorics, number theory, and physics.