Various Aspects Of Multiple Zeta Functions

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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 : 47,6 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 : 49,7 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.

Various Aspects of Multiple Zeta Functions

Hidehiko Mishou,Takashi Nakamura (Mathematician),Masatoshi Suzuki,Yumiko Umegaki
Various Aspects of Multiple Zeta Functions Book PDF

Author : Hidehiko Mishou,Takashi Nakamura (Mathematician),Masatoshi Suzuki,Yumiko Umegaki
Publisher : Advanced Studies in Pure Mathe
Page : 0 pages
File Size : 53,5 Mb
Release : 2020
Category : Mathematics
ISBN : 4864970882

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Various Aspects of Multiple Zeta Functions by Hidehiko Mishou,Takashi Nakamura (Mathematician),Masatoshi Suzuki,Yumiko Umegaki Pdf

This volume is the proceedings of the international conference 'Various Aspects of Multiple Zeta Functions' in honor of Professor Kohji Matsumoto's 60th birthday held at Nagoya University, Japan, during August 21 to 25, 2017.The present volume consists of 15 research papers on various recent topics about multiple zeta-functions, which include not only actually multivariate cases but also single-variable cases, additive and multiplicative number theory, and poly-Bernoulli numbers and polynomials.The editors believe that this volume represents the major part of the contributions presented in the conference, and hope that the volume is useful for all researchers and students who are interested in this fruitful research field.Published by Mathematical Society of Japan and distributed by World Scientific Publishing Co. for all markets except North America

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 : 53,8 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

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 : 41,6 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.

Algebraic and Analytic Aspects of Zeta Function and L―functions

Gautami Bhowmik,Kohji Matsumoto,Hirofumi Tsumura
Algebraic and Analytic Aspects of Zeta Function and L functions Book PDF

Author : Gautami Bhowmik,Kohji Matsumoto,Hirofumi Tsumura
Publisher : Mathematical Society Of Japan Memoirs
Page : 183 pages
File Size : 50,9 Mb
Release : 2010
Category : Functions, Zeta
ISBN : 4931469566

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Algebraic and Analytic Aspects of Zeta Function and L―functions by Gautami Bhowmik,Kohji Matsumoto,Hirofumi Tsumura Pdf

This volume contains lectures presented at the Frenchndash;Japanese Winter School on Zeta and L-functions, held at Muira, Japan, 2008. The main aim of the School was to study various aspects of zeta and L-functions with special emphasis on recent developments. A series of detailed lectures were given by experts in topics that include height zeta-functions, spherical functions and Igusa zeta-functions, multiple zeta values and multiple zeta-functions, classes of Euler products of zeta-functions, and L-functions associated with modular forms. This volume should be helpful to future generations in their study of the fascinating theory of zeta and L-functions. Published by Mathematical Society of Japan and distributed by World Scientific Publishing Co. for all markets

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 : 47,8 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.

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 : 55,5 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.

Series Associated With the Zeta and Related Functions

Hari M. Srivastava,Junesang Choi
Series Associated With the Zeta and Related Functions Book PDF

Author : Hari M. Srivastava,Junesang Choi
Publisher : Springer Science & Business Media
Page : 408 pages
File Size : 48,8 Mb
Release : 2001
Category : Mathematics
ISBN : 0792370546

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Series Associated With the Zeta and Related Functions by Hari M. Srivastava,Junesang Choi Pdf

In recent years there has been an increasing interest in problems involving closed form evaluations of (and representations of the Riemann Zeta function at positive integer arguments as) various families of series associated with the Riemann Zeta function ((s), the Hurwitz Zeta function ((s,a), and their such extensions and generalizations as (for example) Lerch's transcendent (or the Hurwitz-Lerch Zeta function) iI>(z, s, a). Some of these developments have apparently stemmed from an over two-century-old theorem of Christian Goldbach (1690-1764), which was stated in a letter dated 1729 from Goldbach to Daniel Bernoulli (1700-1782), from recent rediscoveries of a fairly rapidly convergent series representation for ((3), which is actually contained in a 1772 paper by Leonhard Euler (1707-1783), and from another known series representation for ((3), which was used by Roger Apery (1916-1994) in 1978 in his celebrated proof of the irrationality of ((3). This book is motivated essentially by the fact that the theories and applications of the various methods and techniques used in dealing with many different families of series associated with the Riemann Zeta function and its aforementioned relatives are to be found so far only"in widely scattered journal articles. Thus our systematic (and unified) presentation of these results on the evaluation and representation of the Zeta and related functions is expected to fill a conspicuous gap in the existing books dealing exclusively with these Zeta functions.

Number Theory

Takashi Aoki,Shigeru Kanemitsu,Jianya Liu
Number Theory Book PDF

Author : Takashi Aoki,Shigeru Kanemitsu,Jianya Liu
Publisher : World Scientific
Page : 267 pages
File Size : 49,5 Mb
Release : 2010
Category : Mathematics
ISBN : 9789814289924

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Number Theory by Takashi Aoki,Shigeru Kanemitsu,Jianya Liu Pdf

This volume aims at collecting survey papers which give broad and enlightening perspectives of various aspects of number theory. Kitaoka''s paper is a continuation of his earlier paper published in the last proceedings and pushes the research forward. Browning''s paper introduces a new direction of research on analytic number theory OCo quantitative theory of some surfaces and Bruedern et al ''s paper details state-of-the-art affairs of additive number theory. There are two papers on modular forms OCo Kohnen''s paper describes generalized modular forms (GMF) which has some applications in conformal field theory, while Liu''s paper is very useful for readers who want to have a quick introduction to Maass forms and some analytic-number-theoretic problems related to them. Matsumoto et al ''s paper gives a very thorough survey on functional relations of root system zeta-functions, HoshiOCoMiyake''s paper is a continuation of Miyake''s long and fruitful research on generic polynomials and gives rise to related Diophantine problems, and Jia''s paper surveys some dynamical aspects of a special arithmetic function connected with the distribution of prime numbers. There are two papers of collections of problems by Shparlinski on exponential and character sums and Schinzel on polynomials which will serve as an aid for finding suitable research problems. Yamamura''s paper is a complete bibliography on determinant expressions for a certain class number and will be useful to researchers. Thus the book gives a good-balance of classical and modern aspects in number theory and will be useful to researchers including enthusiastic graduate students. Sample Chapter(s). Chapter 1: Resent Progress on the Quantitative Arithmetic of Del Pezzo Surfaces (329 KB). Contents: Recent Progress on the Quantitative Arithmetic of Del Pezzo Surfaces (T D Browning); Additive Representation in Thin Sequences, VIII: Diophantine Inequalities in Review (J Brdern et al.); Recent Progress on Dynamics of a Special Arithmetic Function (C-H Jia); Some Diophantine Problems Arising from the Isomorphism Problem of Generic Polynomials (A Hoshi & K Miyake); A Statistical Relation of Roots of a Polynomial in Different Local Fields II (Y Kitaoka); Generalized Modular Functions and Their Fourier Coefficients (W Kohnen); Functional Relations for Zeta-Functions of Root Systems (Y Komori et al.); A Quick Introduction to Maass Forms (J-Y Liu); The Number of Non-Zero Coefficients of a Polynomial-Solved and Unsolved Problems (A Schinzel); Open Problems on Exponential and Character Sums (I E Shparlinski); Errata to OC A General Modular Relation in Analytic Number TheoryOCO (H Tsukada); Bibliography on Determinantal Expressions of Relative Class Numbers of Imaginary Abelian Number Fields (K Yamamura). Readership: Graduate students and researchers in mathematics.

Zeta Regularization Techniques with Applications

E. Elizalde
Zeta Regularization Techniques with Applications Book PDF

Author : E. Elizalde
Publisher : World Scientific
Page : 344 pages
File Size : 55,9 Mb
Release : 1994
Category : Science
ISBN : 9810214413

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Zeta Regularization Techniques with Applications by E. Elizalde Pdf

This book is the result of several years of work by the authors on different aspects of zeta functions and related topics. The aim is twofold. On one hand, a considerable number of useful formulas, essential for dealing with the different aspects of zeta-function regularization (analytic continuation, asymptotic expansions), many of which appear here, in book format, for the first time are presented. On the other hand, the authors show explicitly how to make use of such formulas and techniques in practical applications to physical problems of very different nature. Virtually all types of zeta functions are dealt with in the book.

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 : 674 pages
File Size : 49,8 Mb
Release : 2011-10-11
Category : Mathematics
ISBN : 9780123852199

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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

Multiple Dirichlet Series, Automorphic Forms, and Analytic Number Theory

Bretton Woods Workshop on Multiple Dirichlet Series
Multiple Dirichlet Series Automorphic Forms and Analytic Number Theory Book PDF

Author : Bretton Woods Workshop on Multiple Dirichlet Series
Publisher : American Mathematical Soc.
Page : 320 pages
File Size : 46,7 Mb
Release : 2006
Category : Dirichlet series
ISBN : 9780821839638

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Multiple Dirichlet Series, Automorphic Forms, and Analytic Number Theory by Bretton Woods Workshop on Multiple Dirichlet Series Pdf

Multiple Dirichlet series are Dirichlet series in several complex variables. A multiple Dirichlet series is said to be perfect if it satisfies a finite group of functional equations and has meromorphic continuation everywhere. The earliest examples came from Mellin transforms of metaplectic Eisenstein series and have been intensively studied over the last twenty years. More recently, many other examples have been discovered and it appears that all the classical theorems on moments of $L$-functions as well as the conjectures (such as those predicted by random matrix theory) can now be obtained via the theory of multiple Dirichlet series. Furthermore, new results, not obtainable by other methods, are just coming to light. This volume offers an account of some of the major research to date and the opportunities for the future. It includes an exposition of the main results in the theory of multiple Dirichlet series, and papers on moments of zeta- and $L$-functions, on new examples of multiple Dirichlet

Contributions to the Theory of Zeta-Functions

Shigeru Kanemitsu,Haruo Tsukada
Contributions to the Theory of Zeta Functions Book PDF

Author : Shigeru Kanemitsu,Haruo Tsukada
Publisher : World Scientific
Page : 316 pages
File Size : 52,8 Mb
Release : 2015
Category : Mathematics
ISBN : 9789814449625

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Contributions to the Theory of Zeta-Functions by Shigeru Kanemitsu,Haruo Tsukada Pdf

This volume provides a systematic survey of almost all the equivalent assertions to the functional equations - zeta symmetry - which zeta-functions satisfy, thus streamlining previously published results on zeta-functions. The equivalent relations are given in the form of modular relations in Fox H-function series, which at present include all that have been considered as candidates for ingredients of a series. The results are presented in a clear and simple manner for readers to readily apply without much knowledge of zeta-functions. This volume aims to keep a record of the 150-year-old heritage starting from Riemann on zeta-functions, which are ubiquitous in all mathematical sciences, wherever there is a notion of the norm. It provides almost all possible equivalent relations to the zeta-functions without requiring a reader's deep knowledge on their definitions. This can be an ideal reference book for those studying zeta-functions.

Bernoulli Numbers and Zeta Functions

Tsuneo Arakawa,Tomoyoshi Ibukiyama,Masanobu Kaneko
Bernoulli Numbers and Zeta Functions Book PDF

Author : Tsuneo Arakawa,Tomoyoshi Ibukiyama,Masanobu Kaneko
Publisher : Springer
Page : 274 pages
File Size : 45,5 Mb
Release : 2014-07-11
Category : Mathematics
ISBN : 9784431549192

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Bernoulli Numbers and Zeta Functions by Tsuneo Arakawa,Tomoyoshi Ibukiyama,Masanobu Kaneko Pdf

Two major subjects are treated in this book. The main one is the theory of Bernoulli numbers and the other is the theory of zeta functions. Historically, Bernoulli numbers were introduced to give formulas for the sums of powers of consecutive integers. The real reason that they are indispensable for number theory, however, lies in the fact that special values of the Riemann zeta function can be written by using Bernoulli numbers. This leads to more advanced topics, a number of which are treated in this book: Historical remarks on Bernoulli numbers and the formula for the sum of powers of consecutive integers; a formula for Bernoulli numbers by Stirling numbers; the Clausen–von Staudt theorem on the denominators of Bernoulli numbers; Kummer's congruence between Bernoulli numbers and a related theory of p-adic measures; the Euler–Maclaurin summation formula; the functional equation of the Riemann zeta function and the Dirichlet L functions, and their special values at suitable integers; various formulas of exponential sums expressed by generalized Bernoulli numbers; the relation between ideal classes of orders of quadratic fields and equivalence classes of binary quadratic forms; class number formula for positive definite binary quadratic forms; congruences between some class numbers and Bernoulli numbers; simple zeta functions of prehomogeneous vector spaces; Hurwitz numbers; Barnes multiple zeta functions and their special values; the functional equation of the doub le zeta functions; and poly-Bernoulli numbers. An appendix by Don Zagier on curious and exotic identities for Bernoulli numbers is also supplied. This book will be enjoyable both for amateurs and for professional researchers. Because the logical relations between the chapters are loosely connected, readers can start with any chapter depending on their interests. The expositions of the topics are not always typical, and some parts are completely new.