Multiple zeta functions, multiple polylogarithms and their special values. (English) Zbl 1367.11002

Zhao, Jianqiang

Series on Number Theory and Its Applications 12. Hackensack, NJ: World Scientific (ISBN 978-981-4689-39-7/hbk; 978-981-4689-41-0/ebook). xxi, 595 p. (2016).

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.
Let \(d\in\mathbb{N}\) and \(\underline{s}=(s_1, \ldots, s_d)\in\mathbb{C}\). The multiple zeta function of depth \(d\) is defined by \[ \zeta(\underline{s})= \sum_{k_1> \cdots >k_d>0} k_1^{-s_1}\cdots k_d^{-s_d}, \] \(\mathrm{Re}(s_1+ \cdots + s_j)>j\) 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 \(\zeta^*(\underline{s})\).
Suppose 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 \[ \text{Li}_{s_1, \dots, s_d}(\underline{x})= \sum_{k_1> \cdots > k_d\geq 1} {{x_1^{k_1}\cdots x_d^{k_d}}\over{k_1^{s_1}\cdots k_d^{s_d}}}. \] The multiple zeta values are the convergent special values of the multiple zeta functions at positive integers. Multiple zeta star values are defined similarly.
Let \(\mu=\exp(2\pi \sqrt{-1}/N)\). A colored multiple zeta value of level \(N\) is a number \[ L_n(s_1, \dots, s_d| i_1, \dots, i_d) = \text{Li}_{s_1, \dots, s_d}(\mu^{i_1}, \dots \mu^{i_d}). \]
The book is divided into 15 chapters:
1. Multiple zeta functions.
2. Multiple polylogarithms.
3. Multiple zeta values.
4. Drinfeld associator and single-valued multiple zeta values.
5. Multiple zeta value identities.
6. Symmetrized multiple zeta values.
7. Multiple harmonic sums 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.
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.
Each chapter is accompanied by historical notes and exercises. At the end of the book, 6 appendixes are given, the last of them is devoted to answers to some exercises.
The bibliography contains 644 references.

Reviewer: Renata Macaitiene (Vilnius)

MSC:

11-02	Research exposition (monographs, survey articles) pertaining to number theory
11M32	Multiple Dirichlet series and zeta functions and multizeta values

Keywords:

multiple zeta functions; multiple polylogarithms; colored multiple zeta values