Opera de cribro:
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Providence, Rhode Island
American Mathematical Society
2010
|
| Series: | American Mathematical Society Colloquium publications
volume 57 |
| Subjects: | |
| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020529166&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Physical Description: | XX, 527 Seiten |
| ISBN: | 9780821849705 |
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| 245 | 1 | 0 | |a Opera de cribro |c John Friedlander ; Henryk Iwaniec |
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Record in the Search Index
| _version_ | 1819362060183535616 |
|---|---|
| adam_text | Titel: Opera de cribro
Autor: Friedlander, John B.
Jahr: 2010
Contents
Preface
Chapter 1. Sieve Questions 1
1.1. Inclusion-Exclusion 1
1.2. Some Generality 3
1.3. Some Examples 5
1.4. A Model of a Sifting Sequence for a Given Density 9
Chapter 2. Elementary Considerations on Arithmetic Functions 13
2.1. Dirichlet Convolution and Switching Divisors 13
2.2. Tchebyshev s Prime Sprigs 16
Chapter 3. Bombieri s Sieve 21
3.1. Heuristics for Estimating Sums over Primes 21
3.2. Almost-Primes 23
3.3. Multi-dimensional von Mangoldt Functions 24
3.4. Asymptotic Formula for S(fe) (x) 25
3.5. Application to Additive Convolutions 28
Chapter 4. Sieve of Eratosthenes-Legendre 31
Chapter 5. Sieve Principles and Terminology 35
5.1. Introduction 35
5.2. Cast of Characters 35
5.3. Sifting Weights 38
5.4. Main Terms and Remainders 39
5.5. The Sieve Dimension 42
5.6. Sums of the Local Densities 44
5.7. Sieve-Twisted Sums of the Density Function 45
5.8. Composition of Sieves 49
5.9. Reduced Composition of Sieve-Twisted Sums 50
5.10. An Example of a Reduced Composition 53
Chapter 6. Brun s Sieve - The Big Bang 55
6.1. Brun s Pure Sieve 55
6.2. Setting up a Sieve by Iterations 59
6.3. Compositions using Brun s Sieve 62
6.4. Choosing the Truncation Parameters 64
6.5. Fundamental Lemma 66
6.6. Improved Bounds for the Sifting Limits 71
6.7. Preliminary Sieving 73
vi CONTENTS
6.8. A Cancellation in the Sieve Weights 75
6.9. Applications of Brim s Sieve 77
6.10. Extremely Short Intervals 80
Chapter 7. Selberg s Sieve - Kvadrater er Positive 89
7.1. General Upper-Bound Results 89
7.2. Comments on the A2-Sieve 93
7.3. Crossing Density Functions 98
7.4. Explicit Estimates for J(D, z) 103
7.5. From the Upper-Bound to the Lower-Bound 105
7.6. Selberg s Lower-Bound Sieve Directly 107
7.7. Three Formulas for the Composite Sieve A~A2 112
7.8. Clearing the Sifting Range and a Neat Bound for J(D) 115
7.9. Asymptotic for J(D, z) 117
7.10. Explicit Estimates for the Main Term 118
7.11. Explicit Estimates for the Remainder 120
7.12. Selected Applications 121
7.13. Bounded Gaps Between Primes 122
7.14. Small Gaps Between Primes 133
7.15. Gaps Between Primes and Powers 137
Chapter 8. Sifting Many Residue Classes 139
8.1. Sifting Arbitrary Classes 139
8.2. Sieving for Squares 141
8.3. Ternary Quadratic Forms 144
Chapter 9. The Large Sieve 151
9.1. The Basic Inequality 151
9.2. The Large Sieve Inequality for Additive Characters 154
9.3. Equidistribution over Residue Classes 155
9.4. Arithmetic Large Sieve 157
9.5. Where Linnik meets Selberg: A Duet 159
9.6. The Large Sieve Inequality for Multiplicative Characters 162
9.7. The Larger Sieve of Gallagher 164
9.8. Equidistribution to Large Moduli 166
9.9. Limitations to Equidistribution 171
Chapter 10. Molecular Structure of Sieve Weights 173
10.1. Introduction 173
10.2. What the Upper Bound Leaves Behind 173
10.3. Contribution of Small Prime Factors 175
10.4. Partial Contributions 179
10.5. Related Instances 183
Chapter 11. The Beta-Sieve 185
11.1. Introduction: Buchstab Iterations 185
11.2. Cast of Characters 188
11.3. The Functions F(s), ƒ(s) 190
11.4. The Functions H{s), h(s) for ê 195
11.5. Connections with the Buchstab and Dickman Functions 200
CONTENTS vii
11.6. The Convergence Problem for ê 202
11.7. Estimation of fn(s) for 0 ê 203
11.8. Statement of the Main Theorems of the Beta-Sieve 205
11.9. The Inductive Estimate for Vn(D, z) 207
11.10. Direct Estimation of Vn(D, z) 210
11.11. Completion of the Beta-Sieve for ê 1 212
11.12. Specializing to ê ^ 213
11.13. Estimation of Vn(D,z) 216
11.14. Completion of the Beta-Sieve for 0 ê | 218
11.15. An Extension of the Sifting Range for ê | 218
11.16. Higher Dimension: Composing Beta-Sieves 220
11.17. Estimation of C+{D,z) 221
11.18. Asymptotic Evaluation of Vn(D, z) 223
11.19. Estimates and Numerical Tables for the Sieve Constants 224
11.20. p-adic Zeros of Ternary Quadratic Forms 226
11.21. Representations by Norm Forms 229
11.22. Sums of Two Cubic Norms 231
Chapter 12. The Linear Sieve 235
12.1. A Summary of Previous Results 235
12.2. The True Asymptotics for Special Sifting Functions 237
12.3. The Optimality of the Linear Sieve 240
12.4. A Refinement of Estimates for the Error Terms 240
12.5. Bounds for the Remainder 249
12.6. Linear Sieve with Sharp Error Term 251
12.7. The Remainder in a Well-Factorable Form 252
Chapter 13. Applications to Linear Sequences 259
13.1. Introduction 259
13.2. Large Moduli 261
13.3. Larger Moduli 263
13.4. Short Intervals 267
13.5. Large Primary Factors 271
Chapter 14. The Semi-Linear Sieve 275
14.1. Introduction 275
14.2. A Summary of Previous Results 275
14.3. First Application 277
14.4. Asymptotic Semi-Linear Sieve 278
14.5. Applications of the Asymptotic Semi-Linear Sieve 281
14.6. Shifted Primes as a Sum of Two Squares 282
14.7. Hyperbolic Prime Number Theorem 286
14.8. Prime Points on the Sphere - Secret Challenge 293
Chapter 15. Applications - Choice but not Prime 305
15.1. Squarefree Values of Quadratic Polynomials 305
15.2. Additive Convolution of Multiplicative Functions 310
15.3. Points on Elliptic Curves 315
15.4. Rational Points on a Cubic Surface 322
viii CONTENTS
Chapter 16. Asymptotic Sieve and the Parity Principle 331
16.1. Distribution Functions 331
16.2. The Basic Theorem 332
16.3. Proof of the Basic Theorem 333
16.4. The Parity Phenomenon 337
16.5. The Dichotomy in Action 338
Chapter 17. Combinatorial Identities 345
17.1. Introduction 345
17.2. Sample Combinatorial Identities 346
17.3. An Identity for Composite Numbers 349
17.4. A Further Identity 350
Chapter 18. Asymptotic Sieve for Primes 355
18.1. Combinatorial Transformations 355
18.2. Coupled Exponents of Distribution 359
18.3. Finessing Bombieri s Sieve 360
18.4. Sifting for Primes in Lacunary Sequences 362
Chapter 19. Equidistribution of Quadratic Roots 373
19.1. Equidistribution Modulo One 373
19.2. Equidistribution Modulo Primes 374
19.3. The Poincaré Series 376
19.4. Estimation of Linear Forms 380
19.5. Estimation of Bilinear Forms 381
Chapter 20. Marching over Gaussian Primes 383
20.1. Introduction 383
20.2. Large Sieve for Quadratic Roots 384
20.3. Level of Absolute Distribution 386
20.4. Level of Bilinear Distribution 388
Chapter 21. Primes Represented by Polynomials 395
21.1. Introduction 395
21.2. Prime Values of Quadratic Polynomials in Two Variables 396
21.3. Primes Captured by X2 + Y4 397
21.4. Primes Represented by X3 + 2Y3 399
21.5. A Binary Polynomial of Higher Degree 402
Chapter 22. Level of Distribution of Arithmetic Sequences 405
22.1. Introduction 405
22.2. Primes in Arithmetic Progressions 405
22.3. Absolute Level for Additive Convolutions 409
22.4. Bilinear Level for Quadratic Polynomials 411
22.5. Truncations of Divisor Functions 414
22.6. Gaussian Sequences 417
22.7. Sum of Two Biquadrates 423
22.8. Binary Forms 429
22.9. Binary Cubic Forms with Twists 433
22.10. Level of Distribution Along Orbits 436
CONTENTS ix
22.11. Future Prospects 440
Chapter 23. Primes in Short Intervais 441
23.1. Introduction 441
23.2. The Sieve Argument 443
23.3. Estimates for Dirichlet Polynomials 444
23.4. Triple Products in Short Intervais 450
23.5. Level of Bilinear Distribution 452
Chapter 24. The Least Prime in an Arithmetic Progression 453
24.1. Introduction 453
24.2. The Exceptional Case 454
24.3. A Parity-Preserving Sieve Inequality 457
24.4. Estimation of ipx(x;q,a) 459
24.5. Conclusion 461
24.6. Character Sums over Triple-Primes 461
24.7. Ghosts in the Sieve Opera House 464
Chapter 25. Almost-Prime Sieve 475
25.1. Introduction 475
25.2. Evaluation of the Weighted Sum 476
25.3. Richert s Weights 478
25.4. Almost-Prime Values of Polynomials 480
25.5. Further Applications 481
25.6. Twin Almost-Primes 482
Appendix A. Mean Values of Arithmetic Functions 487
A.I. Simple Estimates 487
A.2. Asymptotic Formulas for Full Sums 491
A.3. Asymptotic Formulas for Restricted Sums 494
A.4. The Linear Case 496
A.5. A Main Term Computation 498
A.6. Evaluation of P(fc)(a;) 501
A.7. Evaluation of/i(fc) (a;) 503
A.8. Congruence Sums 504
Appendix B. Differential-Difference Equations 507
B.I. Adjoint Equations 507
B.2. Zeros of the Adjoint Function 511
B.3. The Largest Zeros of the Adjoint Polynomial 513
B.4. An Example 516
Bibliography 519
Index 525
|
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| author | Friedlander, J. B. 1941- Iwaniec, Henryk 1947- |
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| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
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| dewey-tens | 510 - Mathematics |
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| id | DE-604.BV036608839 |
| illustrated | Not Illustrated |
| indexdate | 2024-12-20T14:38:06Z |
| institution | BVB |
| isbn | 9780821849705 |
| language | English |
| lccn | 2009046518 |
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| physical | XX, 527 Seiten |
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| publisher | American Mathematical Society |
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| series | American Mathematical Society Colloquium publications |
| series2 | American Mathematical Society Colloquium publications |
| spellingShingle | Friedlander, J. B. 1941- Iwaniec, Henryk 1947- Opera de cribro American Mathematical Society Colloquium publications Siebmethode (DE-588)4181206-2 gnd |
| subject_GND | (DE-588)4181206-2 |
| title | Opera de cribro |
| title_auth | Opera de cribro |
| title_exact_search | Opera de cribro |
| title_full | Opera de cribro John Friedlander ; Henryk Iwaniec |
| title_fullStr | Opera de cribro John Friedlander ; Henryk Iwaniec |
| title_full_unstemmed | Opera de cribro John Friedlander ; Henryk Iwaniec |
| title_short | Opera de cribro |
| title_sort | opera de cribro |
| topic | Siebmethode (DE-588)4181206-2 gnd |
| topic_facet | Siebmethode |
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