Multiple zeta functions, multiple polylogarithms and their special values
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) sums.NEWLINENEWLINE10. Multiple zeta star values.NEWLINENEWLINE11. \(q\)-analogs of multiple zeta functions.NEWLINENEWLINE12. \(q\)-analogs of multiple zeta (star) values.NEWLINENEWLINE13. Colored multiple zeta values.NEWLINENEWLINE14. Colored multiple zeta values at lower levels.NEWLINENEWLINE15. Application to Feynman integrals.NEWLINENEWLINEEach 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.NEWLINENEWLINEThe bibliography contains 644 references.
scientific article; zbMATH DE number 6560232