The Theory of Zeta-Functions of Root Systems

Yasushi Komori, Kohji Matsumoto, Hirofumi Tsumura

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English

Springer Verlag, Singapore
03 January 2024

Groups & group theory; Number theory; Complex analysis, complex variables

Series: Springer Monographs in Mathematics

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.

By: Yasushi Komori, Kohji Matsumoto, Hirofumi Tsumura
Imprint: Springer Verlag, Singapore
Country of Publication: Singapore
Edition: 2023 ed.
Dimensions: Height: 235mm, Width: 155mm,
Weight: 799g
ISBN: 9789819909094
ISBN 10: 9819909090
Series: Springer Monographs in Mathematics
Pages: 414
Publication Date: 03 January 2024
Audience: Professional and scholarly , Undergraduate
Format: Hardback
Publisher's Status: Active

Reviews for The Theory of Zeta-Functions of Root Systems

“The book is well structured … . This book can be a good tool for mathematicians interested in multivariable multiple zeta-functions since, together with new results presented and new methods introduced, it provides a necessary basis for the study of them or provide the links to suitable sources.” (Roma Kačinskaitė, zbMATH 1545.11002, 2024)