The Theory of Zeta-Functions of Root Systems

"The Theory of Zeta-Functions of Root Systems" offers a comprehensive examination of the zeta functions of root systems, conceived as a simultaneous generalization of Witten zeta functions, Mordell-Tornheim multiple zeta functions, and Euler-Zagier multiple zeta functions. The authors define zeta functions through specific multiple series formulated in relation to root systems and demonstrate how these intrinsically encompass the actions of the corresponding Weyl groups. The text begins with an introduction to the theory of Lie algebras and root systems, then progresses to the definitions and examples of zeta functions, including meromorphic continuations and recursive analytic structures described by Dynkin diagrams. Special values at integer points as well as functional relationships are also addressed, with the authors highlighting connections to important concepts such as multiple zeta values (MZVs) and their applications in number theory. The book concludes with a brief description of additional relevant topics.