The Theory of Zeta-Functions of Root Systems

Yasushi Komori, Kohji Matsumoto, Hirofumi Tsumura (Autoren)

Buch | Softcover

414 Seiten

2025
Springer Verlag, Singapore
978-981-99-0912-4 (ISBN)

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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.

Introduction.- Fundamentals of the theory of Lie algebras and root systems.- Definitions and examples.- Values at positive even integer points.- Convex polytopes and the rationality.- The recursive structure.- The meromorphic continuation.- Functional relations (I).- Functional relations (II).- Poincar´e polynomials and values at integer points .- The case of the exceptional algebra G2.- Applications to multiple zeta values (I).- Applications to multiple zeta values (II).- L-functions.- Zeta-functions of Lie groups.- Lattice sums of hyperplane arrangements.- Miscellaneous results.

"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 Kacinskait , zbMATH 1545.11002, 2024)

Erscheinungsdatum	05.01.2025
Reihe/Serie	Springer Monographs in Mathematics
Zusatzinfo	13 Illustrations, black and white
Verlagsort	Singapore
Sprache	englisch
Maße	155 x 235 mm
Themenwelt	Mathematik / Informatik ► Mathematik ► Analysis
Themenwelt	Mathematik / Informatik ► Mathematik ► Arithmetik / Zahlentheorie
Schlagworte	Functional relations • Mellin--Barnes Integral Formula • Multiple Zeta-Functions • root systems • Weyl groups
ISBN-10	981-99-0912-0 / 9819909120
ISBN-13	978-981-99-0912-4 / 9789819909124
Zustand	Neuware