Elliptic Functions

Cover......Page 1
Title Page......Page 4
Copyright Page......Page 5
Preface......Page 6
Acknowledgments......Page 8
Contents......Page 10
Notation......Page 14
Introduction......Page 18
Duality and the Differences of Additive Functions......Page 24
First Motive......Page 36
Multiplicative Functions......Page 40
Generalized Turan-Kubilius Inequalities......Page 44
Selberg's Sieve Method......Page 50
Kloosterman Sums......Page 51
CHAPTER 2 A Diophantine Equation......Page 54
CHAPTER 3 A First Upper Bound......Page 70
The First Inductive Proof......Page 81
The Second Inductive Proof......Page 88
Concluding Remarks......Page 93
CHAPTER 4 Intermezzo: The Group Q*/r......Page 95
Duality in Finite Spaces......Page 98
Self-adjoint Maps......Page 100
Duality in Hilbert Space......Page 109
Duality in General......Page 110
Second Motive......Page 114
The Large Sieve and Prime Number Sums......Page 118
The Method of Vinogradov in Vaughan's Form......Page 131
Dirichlet L-Series......Page 136
Additive Functions on Arithmetic Progressions......Page 138
Algebraicanalytic Inequalities......Page 166
CHAPTER 8 The Loop......Page 172
Third Motive......Page 194
CHAPTER 9 The Approximate Functional Equation......Page 200
The Basic Inequality......Page 221
The Decomposition of the Mean......Page 249
Concluding Remarks......Page 256
CHAPTER 11 Some Historical Remarks......Page 261
CHAPTER 12 From L2 to L-......Page 267
CHAPTER 13 A Problem of Katai......Page 276
CHAPTER 14 Inequalities in L-......Page 281
More Duality; Additive Functions as Characters......Page 294
Divisible Groups and Modules......Page 295
Sets of Uniqueness......Page 298
Algorithms......Page 304
CHAPTER 16 The Second Intermezzo......Page 308
A Ring of Operators......Page 314
Practical Measures......Page 324
CHAPTER 18 Simultaneous Product Representations by Values of Rational Functions......Page 326
Linear Recurrences in Modules......Page 327
Elliptic Power Sums......Page 335
Concluding Remarks......Page 345
CHAPTER 19 Simultaneous Product Representations with a;x+b;......Page 346
CHAPTER 20 Information and Arithmetic......Page 360
Transition to Arithmetic......Page 363
Information as an Algebraic Object......Page 370
CHAPTER 21 Central Limit Theorem for Differences......Page 373
CHAPTER 22 Density Theorems......Page 389
Groups of Bounded Order......Page 396
Measures on Dual Groups......Page 397
Arithmic Groups......Page 409
Concluding Remarks......Page 410
Exercises......Page 411
Unsolved Problems......Page 434
Progress in Probabilistic Number Theory......Page 440
Analogues of the Turan-Kubilius Inequality......Page 452
References......Page 466
Subject Index......Page 476