Multiple zeta functions, multiple polylogarithms and their special values
type: Item
instance of: scholarly article
zbMATH Open document ID: 1367.11002
DOI: 10.1142/9634
author: Jianqiang Zhao
publication date: 2016-03-24T00:00:00Z
Mathematics Subject Classification ID: 11-02
Mathematics Subject Classification ID: 11M32
zbMATH DE Number: 6560232
zbMATH Keywords: multiple zeta functions
zbMATH Keywords: multiple polylogarithms
zbMATH Keywords: colored multiple zeta values
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1142/9634
OpenAlex ID: W4230781866
title: Multiple zeta functions, multiple polylogarithms and their special values
published in: Series on Number Theory and its Applications
review text: The book is a very good introduction to the theory of multiple zeta functions and multiple polylogarithms and their special values. It covers the majority of the results in the field since Euler until our days.NEWLINENEWLINELet \(d\in\mathbb{N}\) and \(\underline{s}=(s_1, \ldots, s_d)\in\mathbb{C}\). The multiple zeta function of depth \(d\) is defined by NEWLINE\[NEWLINE \zeta(\underline{s})= \sum_{k_1> \cdots >k_d>0} k_1^{-s_1}\cdots k_d^{-s_d}, NEWLINE\]NEWLINE \(\mathrm{Re}(s_1+ \cdots + s_j)>j\) for all \(j=1, \dots, d\). If the sum runs over \(k_1\geq \cdots \geq k_d>0\), then we have the multiple zeta star function \(\zeta^*(\underline{s})\).NEWLINENEWLINESuppose that \(s_1, \dots, s_d\in \mathbb{N}\) and \(\underline{x} = (x_1, \dots x_d)\in \mathbb{C}^d\) such that \(|\prod_{j=1}^l x_j|<1\) for all \(j=1, \dots, d\). The multiple polylogarithm of depth \(d\) is defined by NEWLINE\[NEWLINE \text{Li}_{s_1, \dots, s_d}(\underline{x})= \sum_{k_1> \cdots > k_d\geq 1} {{x_1^{k_1}\cdots x_d^{k_d}}\over{k_1^{s_1}\cdots k_d^{s_d}}}. NEWLINE\]NEWLINE The multiple zeta values are the convergent special values of the multiple zeta functions at positive integers. Multiple zeta star values are defined similarly.NEWLINENEWLINELet \(\mu=\exp(2\pi \sqrt{-1}/N)\). A colored multiple zeta value of level \(N\) is a number NEWLINE\[NEWLINE L_n(s_1, \dots, s_d| i_1, \dots, i_d) = \text{Li}_{s_1, \dots, s_d}(\mu^{i_1}, \dots \mu^{i_d}). NEWLINE\]NEWLINENEWLINENEWLINEThe book is divided into 15 chapters:NEWLINENEWLINE1. Multiple zeta functions.NEWLINENEWLINE2. Multiple polylogarithms.NEWLINENEWLINE3. Multiple zeta values.NEWLINENEWLINE4. Drinfeld associator and single-valued multiple zeta values.NEWLINENEWLINE5. Multiple zeta value identities.NEWLINENEWLINE6. Symmetrized multiple zeta values.NEWLINENEWLINE7. Multiple harmonic sums and alternating version.NEWLINENEWLINE8. Finite multiple zeta values and finite Euler sums.NEWLINENEWLINE9. \(q\)-analogs of multiple harmonic (star) su…
reviewed by: Renata Macaitienė
description: scientific article; zbMATH DE number 6560232