Value-Distribution of L-Functions

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Springer Science & Business Media, Jun 6, 2007 - Mathematics - 317 pages

These notes present recent results in the value-distribution theory of L-functions with emphasis on the phenomenon of universality. Universality has a strong impact on the zero-distribution: Riemann’s hypothesis is true only if the Riemann zeta-function can approximate itself uniformly. The text proves universality for polynomial Euler products. The authors’ approach follows mainly Bagchi's probabilistic method. Discussion touches on related topics: almost periodicity, density estimates, Nevanlinna theory, and functional independence.

 

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Contents

Introduction
1
12 Bohrs Probabilistic Approach
9
13 Voronins Universality Theorem
12
14 Dirichlet LFunctions and Joint Universality
19
15 LFunctions Associated with Newforms
24
16 The LinnikIbragimov Conjecture
28
Dirichlet Series and Polynomial Euler Products
34
The Main Actors
37
74 Uniqueness Theorems
151
The Riemann Hypothesis
155
82 Bagchis Theorem
156
83 A Generalization
160
84 An Approach Towards Riemanns Hypothesis?
162
85 Further Equivalents of the Riemann Hypothesis
163
Effective Results
167
92 Upper Bounds for the Density of Universality
169

23 Estimates for the Dirichlet Series Coefficients
40
24 The MeanSquare on Vertical Lines
43
Interlude Results from Probability Theory
49
32 Random Elements
52
33 Denjoys Probabilistic Argument for Riemanns Hypothesis
54
34 Characteristic Functions and Fourier Transforms
56
35 Haar Measure and Characters
57
36 Random Processes and Ergodic Theory
58
37 The Space of Analytic Functions
59
Limit Theorems
62
42 Limit Theorems for Dirichlet Polynomials
67
43 An Ergodic Process
70
44 Approximation in the Mean
73
45 A Limit Theorem for Absolutely Convergent Series
76
46 Proof of the Main Limit Theorem
80
47 Generalizations
83
48 A Discrete Limit Theorem
84
Universality
87
52 Application to the Space of Analytic Functions
94
53 Entire Functions of Exponential Type
96
54 The PositiveDensity Method
98
55 The Support of the Limit Measure
105
56 The Universality Theorem
106
57 Discrete Universality
109
The Selberg Class
111
62 Primitive Functions and the Selberg Conjectures
116
63 NonVanishing and Prime Number Theorems
119
64 Pair Correlation
122
65 The PhragménLindelöf Principle
124
66 Universality in the Selberg Class
128
67 Lindelöfs Hypothesis
130
68 Symmetric Power LFunctions
133
ValueDistribution in the Complex Plane
136
72 Riemannvon MangoldtType Formulae
142
73 Nevanlinna Theory
147
93 ValueDistribution on Arithmetic Progressions
174
94 Making Universality Visible
176
95 Almost Periodicity in the HalfPlane of Absolute Convergence
178
96 Effective Inhomogeneous Diophantine Approximation
182
97 cValues Revisited
188
Consequences of Universality
192
102 Functional Independence
195
103 Joint Functional Independence
198
104 Anderssons Disproof of a MeanSquare Conjecture
200
105 Voronins Theorems and Physics
201
106 Shifts of Universal Dirichlet Series
202
Dirichlet Series with Periodic Coefficients
209
112 A Link to the Selberg Class
216
113 Strong Universality
219
114 Hurwitz ZetaFunctions
223
Joint Universality
228
122 A Transfer Theorem
234
123 Twisted LFunctions
238
124 First Applications
245
125 A Conjecture
246
LFunctions of Number Fields
249
132 Grössencharacters
256
133 Hecke LFunctions
258
134 Universality for Hecke LFunctions
260
135 Artins Reciprocity Law
261
136 Artin LFunctions
263
137 The Artin Conjecture
269
138 Joint Universality for Artin LFunctions
274
139 LFunctions to Automorphic Representations
278
Appendix A Short History of Universality
285
References
292
Notations
311
Index
314
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About the author (2007)

Career details of the author:

1996-1999: assistant of Prof. G.J. Rieger at Hanover University

1999: PhD at Hanover University under supervision of Prof. Dr. G.J. Rieger

1999-2004: assistant of Prof. Dr. W. Schwarz and Prof. Dr. J. Wolfart at Frankfurt University

2004: Habilitation at Frankfurt University (venia legendi)

2004-today: 'Ramon y Cajal'-investigador at Universidad Autonoma de Madrid (research fellow)