Duality in Analytic Number Theory

Cover......Page 1
Half Title......Page 2
Title......Page 4
Copyright......Page 5
Table of Contents......Page 6
Acknowledgements......Page 8
Preface......Page 10
Notation......Page 11
Introduction......Page 14
2. Dual spaces......Page 20
3. Spectral decomposition......Page 21
5. Duality principle......Page 22
7. Locally compact groups......Page 24
8. Fourier analysis (mod 1) revisited......Page 25
9. Fourier analysis on R......Page 26
10. Poisson summation......Page 27
12. Characters on Q*......Page 28
13. Fourier analysis on R*......Page 29
14. Automorphic functions......Page 30
1 Background philosophy......Page 35
2 Operator norm inequalities......Page 37
3 Dual norm inequalities......Page 44
4 Exercises: Including the Large Sieve......Page 51
Characters on finite abelian groups......Page 53
The Large Sieve via the duality principle and Poisson summation......Page 55
Why is the Large Sieve a sieve?......Page 57
Around the Selberg sieve......Page 58
5 The method of the stable dual:
 Deriving the approximate functional equations......Page 67
6 The method of the stable dual:
 Solving the approximate functional equations......Page 71
7 Exercises: Almost linear, almost exponential......Page 87
8 Additive functions of class Lα  .   A first application of the method......Page 98
9 Multiplicative functions of the class Lα:  First Approach......Page 103
10 Multiplicative functions of the class Lα:  Second Approach......Page 112
11 Multiplicative functions of the class Lα:  Third Approach......Page 120
Remarks......Page 124
A little topology......Page 130
Almost periodic functions......Page 131
13 Theorems of Wirsing and Halász......Page 134
14 Again Wirsing\'s theorem......Page 141
15 Exercises: The prime number theorem......Page 146
16 Finitely distributed additive functions......Page 152
17 Multiplicative functions of the class Lα .  Mean value zero......Page 158
18 Exercises: Including logarithmic weights......Page 167
Weighted sums; Logarithmic density......Page 168
19 Encounters with Ramanujan\'s function r(n)......Page 170
20 The operator T on L2......Page 178
21 The operator T on Lα  and other spaces......Page 188
The operator D and differentiation......Page 202
The operator T and the convergence of measures......Page 203
23 Pause: Towards the discrete derivative......Page 209
Young-Hausdorff inequalities on a finite group. Riesz-Thorin the-\rorem......Page 217
24 Exercises: Multiplicative functions on arithmetic progressions; Wiener phenomenon......Page 224
Wiener phenomenon on finite groups......Page 228
25 Fractional power Large Sieves.
Operators involving primes......Page 230
Operators involving primes......Page 241
26 Exercises: Probability seen from number theory......Page 251
27 Additive functions on arithmetic progressions: 
Small moduli......Page 254
28 Additive functions on arithmetic progressions: 
Large moduli......Page 258
With additive characters......Page 273
With multiplicative characters......Page 274
Again without characters......Page 279
Maximal versions of Theorem 28.1, case α = 2......Page 280
Maximal version of Theorem 28.1, case 1 < α  < 2......Page 284
30 Shift operators and orthogonal duals......Page 290
31 Differences of additive functions. 
Local inequalities......Page 294
32 Linear forms in shifted additive functions......Page 304
A ring of operators......Page 305
33 Exercises: Stability;
 Correlations of multiplicative functions......Page 314
Norms of compositions of arithmetic operators......Page 321
Multiplicative functions and almost periodicity......Page 323
Arithmetic functions on short intervals......Page 325
The correlation of multiplicative functions and sums of additive functions......Page 334
Rings of shift operators......Page 336
Abstract semigroups......Page 337
35 Rückblick......Page 339
References......Page 340
Author Index......Page 352
Subject Index......Page 354