Multiple Zeta Functions, Multiple Polylogarithms and Their Special Values

Description

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 Contents Preface Introduction 1 Multiple Zeta Functions 2 Multiple Polylogarithms (MPLs) 3 Multiple Zeta Values (MZVs) 4 Drinfeld Associator and Single-Valued MZVs 5 Multiple Zeta Value Identities 6 Symmetrized Multiple Zeta Values (SMZVs) 7 Multiple Harmonic Sums (MHSs) and Alternating Version 8 Finite Multiple Zeta Values and Finite Euler Sums 9 q-Analogs of Multiple Harmonic (Star) Sums 10 Multiple Zeta Star Values (MZ⋆Vs) 11 q-Analogs of Multiple Zeta Functions 12 q-Analogs of Multiple Zeta (Star) Values 13 Colored Multiple Zeta Values 14 Colored Multiple Zeta Values at Lower Levels 15 Application to Feynman Integrals A - Key Concepts of Hopf Algebras B - Some Useful Results from Lie Algebras C - Basics of Hypergeometric Functions D - Sample Computer Codes E - Tables of Special Values F - Answers to Some Exercises Bibliography List of Abbreviations List of Symbols Index