The Theory of Zeta-Functions of Root Systems

Preface
Contents
Chapter 1 Introduction
	1.1. The Euler double zeta-function
	1.2. Multiple zeta-functions of Barnes and of Mellin
	1.3. Euler–Zagier multiple zeta-functions
	1.4. Mordell–Tornheim multiple zeta-functions
	1.5. Witten multiple zeta-functions
	1.6. Zeta-functions of root systems
Chapter 2 Fundamentals of the theory of Lie algebras and root systems
	2.1. Lie algebras
	2.2. Roots
	2.3. Abstract root systems
	2.4. Weights
Chapter 3 Definitions and examples
	3.1. The definition of zeta-functions of root systems
	3.2. Explicit forms for zeta-functions of type A
	3.3. Explicit forms for zeta-functions of other types
Chapter 4 Values at positive even integer points
	4.1. The sum S(s,y;)
	4.2. The values at even integer points
	4.3. A closed expression of F(t,y;) and further examples
Chapter 5 Convex polytopes and the rationality
	5.1. Convex polytopes
	5.2. The rationality
Chapter 6 The recursive structure
	6.1. The Mellin–Barnes integral formula
	6.2. The recursive structure for Ar
	6.3. The recursive structure for Br, Cr and Dr
	6.4. Recursive structures and Dynkin diagrams
Chapter 7 The meromorphic continuation
	7.1. The meromorphic continuation of zeta-functions of root systems
	7.2. Multiple zeta-functions defined by linear forms
Chapter 8 Functional relations (I)
	8.1. A method to evaluate the Riemann zeta-function
	8.2. Functional relations for the zeta-function of A2
	8.3. Functional relations for the zeta-function of C2
	8.4. An application of Nakamura\'s method to the zeta-function of G2
Chapter 9 Functional relations (II)
	9.1. A sketch in the case of A2
	9.2. Lemmas on root systems and Weyl groups
	9.3. Automorphisms on Dynkin diagrams
	9.4. General functional relations
	9.5. An extension
	9.6. The action of `3́9`42`\"̇613A``45`47`\"603AAut()
	9.7. Explicit generating functions (# I=(r-1) case)
	9.8. Explicit functional relations (# I=(r-1) case)
	9.9. Explicit functional relations (# I=1 case)
Chapter 10 Poincaré polynomials and values at integer points
	10.1. Poincaré polynomials
	10.2. The non-vanishing of the sum of coefficients
Chapter 11 The case of the exceptional algebra G2
	11.1. A criterion of reduction
	11.2. Application of Theorem 11.1 to the case G2
	11.3. Some lemmas
	11.4. A functional relation corresponding to I={2}
Chapter 12 Applications to multiple zeta values (I)
	12.1. The case of type A2
	12.2. The case of type Cr
	12.3. The case of type Br
	12.4. Restricted sum formulas
	12.5. Parity results
Chapter 13 Applications to multiple zeta values (II)
	13.1. Hoffman\'s algebra
	13.2. Extension of Hoffman\'s setup
	13.3. Double and triple zeta values
	13.4. Proofs of results in Section 13.2
	13.5. Functional relations including double shuffle relations
Chapter 14 L-functions
	14.1. L-functions of root systems
	14.2. Proofs of theorems
	14.3. Special values of L-functions
Chapter 15 Zeta-functions of Lie groups
	15.1. The background and the motivation
	15.2. A general form of zeta-functions
	15.3. Zeta-functions of weight lattices of Lie groups
	15.4. Explicit forms of zeta-functions
	15.5. An application to partial multiple zeta values
	15.6. Functional relations and various evaluation formulas
	15.7. Parity results
Chapter 16 Lattice sums of hyperplane arrangements
	16.1. Notations and the statement of results on lattice sums
	16.2. Examples
	16.3. Proof of Theorem 16.2
	16.4. The structure of the proof of Theorems 16.4 and 16.5
	16.5. The generating function and convex polytopes
	16.6. Properties of the polytopes P(m;y)
	16.7. Completion of the proof of Theorems 16.4 and 16.5
	16.8. A hierarchy and differential equations
	16.9. Proof of Theorems 9.10, 9.11 and 9.12
	16.10. Proof of Theorem 9.14
Chapter 17 Miscellaneous results
	17.1. Functional equations
	17.2. The desingularization
	17.3. The number of representations of SU(3)
	17.4. A connection with Schur multiple zeta-functions
	17.5. Recent results on functional relations
Bibliography
Index