Introduction / 1: |
A Brief History of Zeta Functions / 1.1: |
Euler, Riemann / 1.1.1: |
Dirichlet / 1.1.2: |
Dedekind / 1.1.3: |
Artin, Weil / 1.1.4: |
Birch, Swinnerton-Dyer / 1.1.5: |
Zeta Functions of Groups / 1.2: |
Zeta Functions of Algebraic Groups / 1.2.1: |
Zeta Functions of Rings / 1.2.2: |
Local Functional Equations / 1.2.3: |
Uniformity / 1.2.4: |
Analytic Properties / 1.2.5: |
p-Adic Integrals / 1.3: |
Natural Boundaries of Euler Products / 1.4: |
Nilpotent Groups: Explicit Examples / 2: |
Calculating Zeta Functions of Groups / 2.1: |
Calculating Zeta Functions of Lie Rings / 2.2: |
Constructing the Cone Integral / 2.2.1: |
Resolution / 2.2.2: |
Evaluating Monomial Integrals / 2.2.3: |
Summing the Rational Functions / 2.2.4: |
Explicit Examples / 2.3: |
Free Abelian Lie Rings / 2.4: |
Heisenberg Lie Ring and Variants / 2.5: |
Grenham's Lie Rings / 2.6: |
Free Class-2 Nilpotent Lie Rings / 2.7: |
Three Generators / 2.7.1: |
n Generators / 2.7.2: |
The 'Elliptic Curve Example' / 2.8: |
Other Class Two Examples / 2.9: |
The Maximal Class Lie Ring M[subscript 3] and Variants / 2.10: |
Lie Rings with Large Abelian Ideals / 2.11: |
F[subscript 3,2] / 2.12: |
The Maximal Class Lie Rings M[subscript 4] and Fil[subscript 4] / 2.13: |
Nilpotent Lie Algebras of Dimension [less than or equal] 6 / 2.14: |
Nilpotent Lie Algebras of Dimension 7 / 2.15: |
Soluble Lie Rings / 3: |
Proof of Theorem 3.1 / 3.1: |
Choosing a Basis for tr[subscript n](Z) / 3.2.1: |
Determining the Conditions / 3.2.2: |
Constructing the Zeta Function / 3.2.3: |
Transforming the Conditions / 3.2.4: |
Deducing the Functional Equation / 3.2.5: |
Variations / 3.3: |
Quotients of tr[subscript n](Z) / 3.4.1: |
Counting All Subrings / 3.4.2: |
Algebraic Groups / 4: |
Nilpotent Groups and Lie Rings / 4.3: |
The Conjecture / 4.4: |
Special Cases Known to Hold / 4.5: |
A Special Case of the Conjecture / 4.6: |
Projectivisation / 4.6.1: |
Manipulating the Cone Sums / 4.6.2: |
Cones and Schemes / 4.6.4: |
Quasi-Good Sets / 4.6.5: |
Quasi-Good Sets: The Monomial Case / 4.6.6: |
Applications of Conjecture 4.5 / 4.7: |
Counting Subrings and p-Subrings / 4.8: |
Counting Ideals and p-Ideals / 4.9: |
Heights, Cocentral Bases and the [pi]-Map / 4.9.1: |
Property ([dagger]) / 4.9.2: |
Lie Rings Without ([dagger]) / 4.9.3: |
Natural Boundaries I: Theory / 5: |
A Natural Boundary for [zeta]GSp[subscript 6] (s) / 5.1: |
Natural Boundaries for Euler Products / 5.2: |
Practicalities / 5.2.1: |
Distinguishing Types I, II and III / 5.2.2: |
Avoiding the Riemann Hypothesis / 5.3: |
All Local Zeros on or to the Left of R(s) = [beta] / 5.4: |
Using Riemann Zeros / 5.4.1: |
Avoiding Rational Independence of Riemann Zeros / 5.4.2: |
Continuation with Finitely Many Riemann Zeta Functions / 5.4.3: |
Infinite Products of Riemann Zeta Functions / 5.4.4: |
Natural Boundaries II: Algebraic Groups / 6: |
G = GO[subscript 2l+1] of Type B[subscript l] / 6.1: |
G = GSp[subscript 2l] of Type C[subscript l] or G = GO[superscript +][subscript 2l] of Type D[subscript l] / 6.3: |
G = GSp[subscript 2l] of Type C[subscript l] / 6.3.1: |
G = GO[superscript + subscript 2l] of Type D[subscript l] / 6.3.2: |
Natural Boundaries III: Nilpotent Groups / 7: |
Zeta Functions with Meromorphic Continuation / 7.1: |
Zeta Functions with Natural Boundaries / 7.3: |
Type I / 7.3.1: |
Type II / 7.3.2: |
Type III / 7.3.3: |
Other Types / 7.4: |
Types IIIa and IIIb / 7.4.1: |
Types IV, V and VI / 7.4.2: |
Large Polynomials / A: |
H[superscript 4], Counting Ideals / A.1: |
g[subscript 6,4], Counting All Subrings / A.2: |
T[subscript 4], Counting All Subrings / A.3: |
L[subscript (3,2,2)], Counting Ideals / A.4: |
G[subscript 3] x g[subscript 5,3], Counting Ideals / A.5: |
g[subscript 6,12], Counting All Subrings / A.6: |
g[subscript 1357G], Counting Ideals / A.7: |
g[subscript 1457A], Counting Ideals / A.8: |
g[subscript 1457B], Counting Ideals / A.9: |
tr[subscript 6](Z), Counting Ideals / A.10: |
tr[subscript 7](Z), Counting Ideals / A.11: |
Factorisation of Polynomials Associated to Classical Groups / B: |
References |
Index |
Index of Notation |
Introduction / 1: |
A Brief History of Zeta Functions / 1.1: |
Euler, Riemann / 1.1.1: |
Dirichlet / 1.1.2: |
Dedekind / 1.1.3: |
Artin, Weil / 1.1.4: |