Zeta Functions over Zeros of Zeta Functions (eBook)

Preface 7
Contents 10
List of Special Symbols 14
1 Introduction 16
1.1 Symmetric Functions 16
1.2 Essential Basic Notation 18
1.3 The Poisson Summation Formula 19
1.4 Euler–Maclaurin Summation Formulae 20
1.5 Meromorphic Properties of Mellin Transforms 21
2 Infinite Products and Zeta-Regularization 24
2.1 Informal Discussion 24
2.2 A Class of Eligible Sequences {xk} 27
2.3 Meromorphic Continuation of the Zeta Function 27
2.4 The Generalized Zeta Function 29
2.5 The Zeta-Regularized Product 30
2.6 Practical Results 33
2.6.1 Zeta-Regularization: a Zeta-Free Recipe 33
2.6.2 A Subclass: ``Theta-Eligible" Sequences 34
2.6.3 Explicit Properties of the Generalized Zeta Function 36
3 The Riemann Zeta Function (x): a Primer 38
3.1 Definition and Immediate Properties 38
3.2 The Euler Infinite Product 39
3.3 The Stieltjes and Cumulant Expansions 39
3.4 The Functional Equation and Completed Zeta Function (x) 40
3.5 The Dirichlet Beta Function (x) 43
3.6 The Hurwitz Zeta Function (x,w) 43
4 Riemann Zeros and Factorizations of the Zeta Function 47
4.1 Growth Properties of (x) and (x) 47
4.2 The Riemann Zeros (Basic Features) 48
4.3 Hadamard Products for (x) 49
4.4 Basic Bounds on '/ 50
4.5 The (Asymptotic) Riemann–von Mangoldt Formula 53
5 Superzeta Functions: an Overview 55
5.1 First Kind (Z) 56
5.2 Second Kind (Z) 56
5.3 Third Kind (Z) 58
5.4 Further Generalizations (Lerch, Cramér, …) 58
5.5 Other Studies on Superzeta Functions 59
6 Explicit Formulae 62
6.1 The Guinand–Weil Explicit Formula 62
6.2 Derivation of the Explicit Formula 63
6.3 Pattern-Matching with the Selberg Trace Formula 65
6.3.1 The Selberg Trace Formula (Compact Surface Case) 65
6.3.2 Comparison with the Explicit Formula 66
6.4 Explicit Formulae for the Superzeta Functions 67
6.4.1 The Family of the First Kind Z 67
6.4.2 The Family of the Second Kind Z 69
6.4.3 Concluding Remarks 71
7 The Family of the First Kind {Z(s |t)} 72
7.1 The Basic Analytical Continuation Formula 72
7.2 Derivations 73
7.2.1 Derivation by Contour Integration 73
7.2.2 Derivation by Eligibility of the Riemann Zeros 75
7.3 Analytic Properties of the Family { Z(s |t) } 77
7.4 Special Values of Z(s |t) for General t 78
7.5 Imprints of the Central Symmetry -3mu(1-) 80
7.5.1 t -3mu(-t) Symmetry at Integer t 80
7.5.2 Sum Rules at an Arbitrary Fixed t 81
7.6 Special Values of Z(s |t) at t=0 and 1 2 82
7.6.1 The Function Z0(s) (the Confluent Case t=0) 82
7.6.2 The Function Z(s) (the Case t=1 2) 83
7.7 Tables of Formulae for the Special Values of Z 84
7.7.1 Function of First Kind for General t 85
7.7.2 Function of First Kind at t=0 and 1 2 86
8 The Family of the Second Kind { Z(|t) } 87
8.1 The Confluent Case Z(|t=0) Z0() 88
8.2 Meromorphic Continuation in for General t 89
8.3 Algebraic Results for Z(|t) at General t 91
8.4 Transcendental Values of Z(|t) for General t 92
8.5 Imprints of the Central Symmetry -3mu(1-) 92
8.6 Results for Z(|t) at t=0 and t=1 2 94
8.6.1 The Function Z0(s) (the Confluent Case t=0) 94
8.6.2 The Function Z(s) (the Case t=1 2) 94
8.7 Tables of Formulae for the Special Values of Z 96
8.7.1 Function of Second Kind for General t 96
8.7.2 Function of Second Kind at t=0 and 1 2 97
9 The Family of the Third Kind { Z(s |)} 98
10 Extension to Other Zeta- and L-Functions 101
10.1 Admissible Primary Functions L(x) 102
10.2 The Three Superzeta Families 103
10.3 The First Family { Z} 104
10.3.1 The Zeta Function Z (s |t) over the Trivial Zeros 104
10.3.2 The Basic Analytical Continuation Formula for Z 105
10.3.3 Special Values of Z(s |t) for General t 106
10.3.4 Special Values of Z(s |t) at t=0 and 1 2 107
10.4 The Second Family { Z} 108
10.4.1 The Confluent Case Z(|t=0) Z0() 108
10.4.2 Algebraic Results for Z(|t) at General t 108
10.4.3 Transcendental Values of Z(|t) at General t 109
10.5 The Third Family { Z} 110
10.6 Special Concrete Examples 111
10.6.1 L-Functions of Real Primitive Dirichlet Characters 111
10.6.2 Dedekind Zeta Functions 115
10.7 Tables of Formulae for the Special Values 118
10.7.1 For General Primary Functions L(x) at General t 119
10.7.2 Dirichlet-L Cases, Functions of First Kind at t=0 and 1 2 120
10.7.3 Dedekind- Cases, Functions of First Kind at t=0 and 1 2 121
11 Application: an Asymptotic Criterion for the Riemann Hypothesis 122
11.1 Introduction to the Result 122
11.2 Asymptotic Alternative for n, n 124
11.2.1 The Case [RH False] 125
11.2.2 The Case [RH True] 126
11.2.3 Recapitulation and Discussion 127
11.3 An Even More Sensitive Sequence 128
11.4 More General Cases: a Summary 130
A Numerical Explorations 131
A.1 Superzeta Functions of the Second Kind 131
A.2 Superzeta Functions of the First Kind 133
A.3 Numerical Tables 134
B The Selberg Case 136
B.1 Superzeta Functions of the First Kind 137
B.2 Superzeta Functions of the Second Kind 139
B.3 Tables of Special-Value Formulae (Selberg Cases) 141
C On the Logarithmic Derivatives at 1 2 144
D On the Zeros of the Zeta Function by Hj. Mellin (1917) 146
References 160
Index 166