The famous zeros of the Riemann zeta function and its generalizations (L-functions, Dedekind and Selberg zeta functions) are analyzed through several zeta functions built over those zeros. These ‘second-generation’ zeta functions have surprisingly many explicit, yet largely unnoticed properties, which are surveyed here in an accessible and synthetic manner, and then compiled in numerous tables. No previous book has addressed this neglected topic in analytic number theory. Concretely, this handbook will help anyone faced with symmetric sums over zeros like Riemann’s. More generally, it aims at reviving the interest of number theorists and complex analysts toward those unfamiliar functions, on the 150th anniversary of Riemann’s work.
Author(s): André Voros
Series: Lecture Notes Of The Unione Matematica Italiana
Edition: 1st Edition.
Publisher: Springer
Year: 2009
Language: English
Pages: 171
Tags: Математика;Теория чисел;
3642052029......Page 1
Zeta Functions
over Zeros of
Zeta Functions
......Page 2
Preface......Page 7
List of Special Symbols......Page 14
1.1 Symmetric Functions......Page 16
1.2 Essential Basic Notation......Page 18
1.3 The Poisson Summation Formula......Page 19
1.4 Euler–Maclaurin Summation Formulae......Page 20
1.5 Meromorphic Properties of Mellin Transforms......Page 21
2.1 Informal Discussion......Page 24
2.3 Meromorphic Continuation of the Zeta Function......Page 27
2.4 The Generalized Zeta Function......Page 29
2.5 The Zeta-Regularized Product......Page 30
2.6.1 Zeta-Regularization: a Zeta-Free Recipe......Page 33
2.6.2 A Subclass: ``Theta-Eligible" Sequences......Page 34
2.6.3 Explicit Properties of the Generalized Zeta Function......Page 36
3.1 Definition and Immediate Properties......Page 38
3.3 The Stieltjes and Cumulant Expansions......Page 39
3.4 The Functional Equation and Completed Zeta Function (x)......Page 40
3.6 The Hurwitz Zeta Function (x,w)......Page 43
4.1 Growth Properties of (x) and (x)......Page 47
4.2 The Riemann Zeros (Basic Features)......Page 48
4.3 Hadamard Products for (x)......Page 49
4.4 Basic Bounds on '/......Page 50
4.5 The (Asymptotic) Riemann–von Mangoldt Formula......Page 53
5 Superzeta Functions: an Overview......Page 55
5.2 Second Kind (Z)......Page 56
5.4 Further Generalizations (Lerch, Cramér, …)......Page 58
5.5 Other Studies on Superzeta Functions......Page 59
6.1 The Guinand–Weil Explicit Formula......Page 62
6.2 Derivation of the Explicit Formula......Page 63
6.3.1 The Selberg Trace Formula (Compact Surface Case)......Page 65
6.3.2 Comparison with the Explicit Formula......Page 66
6.4.1 The Family of the First Kind Z......Page 67
6.4.2 The Family of the Second Kind Z......Page 69
6.4.3 Concluding Remarks......Page 71
7.1 The Basic Analytical Continuation Formula......Page 72
7.2.1 Derivation by Contour Integration......Page 73
7.2.2 Derivation by Eligibility of the Riemann Zeros......Page 75
7.3 Analytic Properties of the Family { Z(s |t) }......Page 77
7.4 Special Values of Z(s |t) for General t......Page 78
7.5.1 t -3mu(-t) Symmetry at Integer t......Page 80
7.5.2 Sum Rules at an Arbitrary Fixed t......Page 81
7.6.1 The Function Z0(s) (the Confluent Case t=0)......Page 82
7.6.2 The Function Z(s) (the Case t=1 2)......Page 83
7.7 Tables of Formulae for the Special Values of Z......Page 84
7.7.1 Function of First Kind for General t......Page 85
7.7.2 Function of First Kind at t=0 and 1 2......Page 86
8 The Family of the Second Kind { Z(|t) }......Page 87
8.1 The Confluent Case Z(|t=0) Z0()......Page 88
8.2 Meromorphic Continuation in for General t......Page 89
8.3 Algebraic Results for Z(|t) at General t......Page 91
8.5 Imprints of the Central Symmetry -3mu(1-)......Page 92
8.6.2 The Function Z(s) (the Case t=1 2)......Page 94
8.7.1 Function of Second Kind for General t......Page 96
8.7.2 Function of Second Kind at t=0 and 1 2......Page 97
9 The Family of the Third Kind { Z(s |)}......Page 98
10 Extension to Other Zeta- and L-Functions......Page 101
10.1 Admissible Primary Functions L(x)......Page 102
10.2 The Three Superzeta Families......Page 103
10.3.1 The Zeta Function Z (s |t) over the Trivial Zeros......Page 104
10.3.2 The Basic Analytical Continuation Formula for Z......Page 105
10.3.3 Special Values of Z(s |t) for General t......Page 106
10.3.4 Special Values of Z(s |t) at t=0 and 1 2......Page 107
10.4.2 Algebraic Results for Z(|t) at General t......Page 108
10.4.3 Transcendental Values of Z(|t) at General t......Page 109
10.5 The Third Family { Z}......Page 110
10.6.1 L-Functions of Real Primitive Dirichlet Characters......Page 111
10.6.2 Dedekind Zeta Functions......Page 115
10.7 Tables of Formulae for the Special Values......Page 118
10.7.1 For General Primary Functions L(x) at General t......Page 119
10.7.2 Dirichlet-L Cases, Functions of First Kind at t=0 and 1 2......Page 120
10.7.3 Dedekind- Cases, Functions of First Kind at t=0 and 1 2......Page 121
11.1 Introduction to the Result......Page 122
11.2 Asymptotic Alternative for n, n......Page 124
11.2.1 The Case [RH False]......Page 125
11.2.2 The Case [RH True]......Page 126
11.2.3 Recapitulation and Discussion......Page 127
11.3 An Even More Sensitive Sequence......Page 128
11.4 More General Cases: a Summary......Page 130
Appendix......Page 0
A.1 Superzeta Functions of the Second Kind......Page 131
A.2 Superzeta Functions of the First Kind......Page 133
A.3 Numerical Tables......Page 134
B The Selberg Case......Page 136
B.1 Superzeta Functions of the First Kind......Page 137
B.2 Superzeta Functions of the Second Kind......Page 139
B.3 Tables of Special-Value Formulae (Selberg Cases)......Page 141
C On the Logarithmic Derivatives at 1 2......Page 144
D On the Zeros of the Zeta Function by Hj. Mellin (1917)......Page 146
References......Page 160
Index......Page 166
Editorial Policy
......Page 171
