Problems in analytic number theory:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York [u.a.]
Springer
2001
|
| Series: | Graduate texts in mathematics
206 : Readings in mathematics |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVI, 452 S. 25 cm |
| ISBN: | 0387951431 |
Staff View
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Record in the Search Index
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| adam_text | Contents
Preface vii
Acknowledgments xi
I Problems 1
1 Arithmetic Functions 3
1.1 The Mobius Inversion Formula and Applications . . 4
1.2 Formal Dirichlet Series 7
1.3 Orders of Some Arithmetical Functions 9
1.4 Average Orders of Arithmetical Functions 10
1.5 Supplementary Problems 11
2 Primes in Arithmetic Progressions 17
2.1 Summation Techniques 17
2.2 Characters mod q 22
2.3 Dirichlet s Theorem 24
2.4 Dirichlet s Hyperbola Method 27
2.5 Supplementary Problems 29
xiv Contents
3 The Prime Number Theorem 35
3.1 Chebyshev s Theorem 36
3.2 Nonvanishing of Dirichlet Series on Re(s) = 1 .... 39
3.3 The Ikehara Wiener Theorem 42
3.4 Supplementary Problems 48
4 The Method of Contour Integration 53
4.1 Some Basic Integrals 53
4.2 The Prime Number Theorem 57
4.3 Further Examples 62
4.4 Supplementary Problems 64
5 Functional Equations 69
5.1 Poisson s Summation Formula 69
5.2 The Riemann Zeta Function 72
5.3 Gauss Sums 75
5.4 Dirichlet L functions 76
5.5 Supplementary Problems 79
6 Hadamard Products 85
6.1 Jensen s Theorem 85
6.2 Entire Functions of Order 1 88
6.3 The Gamma Function 91
6.4 Infinite Products for £(s) and £(s, x) 93
6.5 Zero Free Regions for ((s) and L(s,x) 94
6.6 Supplementary Problems 99
7 Explicit Formulas 101
7.1 Counting Zeros 101
7.2 Explicit Formula for ip{x) 104
7.3 Weil s Explicit Formula 107
7.4 Supplementary Problems Ill
8 The Selberg Class 115
8.1 The Phragmen Lindelof Theorem 116
8.2 Basic Properties 118
8.3 Selberg s Conjectures 123
8.4 Supplementary Problems 125
Contents xv
9 Sieve Methods 127
9.1 The Sieve of Eratosthenes 127
9.2 Brun s Elementary Sieve 133
9.3 Selberg s Sieve 138
9.4 Supplementary Problems 143
10 p adic Methods 147
10.1 Ostrowski s Theorem 147
10.2 Hensel s Lemma 155
10.3 p adic Interpolation 159
10.4 The p adic Zeta Function 165
10.5 Supplementary Problems 168
11 Solutions 171
1 Arithmetic Functions 173
1.1 The Mobius Inversion Formula and Applications . . 173
1.2 Formal Dirichlet Series 182
1.3 Orders of Some Arithmetical Functions 186
1.4 Average Orders of Arithmetical Functions 189
1.5 Supplementary Problems 194
2 Primes in Arithmetic Progressions 211
2.1 Characters mod q 211
2.2 Dirichlet s Theorem 220
2.3 Dirichlet s Hyperbola Method 225
2.4 Supplementary Problems 231
3 The Prime Number Theorem 247
3.1 Chebyshev s Theorem 247
3.2 Nonvanishing of Dirichlet Series on Re(s) = 1 .... 254
3.3 The Ikehara Wiener Theorem 262
3.4 Supplementary Problems 266
4 The Method of Contour Integration 279
4.1 Some Basic Integrals 279
4.2 The Prime Number Theorem 285
4.3 Further Examples 288
4.4 Supplementary Problems 291
xvi Contents
5 Functional Equations 303
5.1 Poisson s Summation Formula 303
5.2 The Riemann Zeta Function 306
5.3 Gauss Sums 307
5.4 Dirichlet L functions 309
5.5 Supplementary Problems 312
6 Hadamard Products 331
6.1 Jensen s theorem 331
6.2 The Gamma Function 332
6.3 Infinite Products for £(s) and £{s,x) 343
6.4 Zero Free Regions for ((s) and L(s,x) 348
6.5 Supplementary Problems 353
7 Explicit Formulas 357
7.1 Counting Zeros 357
7.2 Explicit Formula for ip{x) 360
7.3 Supplementary Problems 366
8 The Selberg Class 377
8.1 The Phragmen Lindelof Theorem 377
8.2 Basic Properties 378
8.3 Selberg s Conjectures 385
8.4 Supplementary Problems 390
9 Sieve Methods 397
9.1 The Sieve of Eratosthenes 397
9.2 Brun s Elementary Sieve 404
9.3 Selberg s Sieve 406
9.4 Supplementary Problems 413
10 p adic Methods 423
10.1 Ostrowski s Theorem 423
10.2 Hensel s Lemma 428
10.3 p adic Interpolation 431
10.4 The p adic ( Function 437
10.5 Supplementary Problems 441
References 447
Index 449
|
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| indexdate | 2024-07-09T18:49:28Z |
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| isbn | 0387951431 |
| language | English |
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| physical | XVI, 452 S. 25 cm |
| publishDate | 2001 |
| publishDateSearch | 2001 |
| publishDateSort | 2001 |
| publisher | Springer |
| record_format | marc |
| series | Graduate texts in mathematics |
| series2 | Graduate texts in mathematics |
| spelling | Murty, Maruti Ram 1953- Verfasser (DE-588)120837307 aut Problems in analytic number theory M. Ram Murty New York [u.a.] Springer 2001 XVI, 452 S. 25 cm txt rdacontent n rdamedia nc rdacarrier Graduate texts in mathematics 206 : Readings in mathematics Getaltheorie gtt Nombres, Théorie des - Problèmes et exercices Problemen gtt Number theory Analytische Zahlentheorie (DE-588)4001870-2 gnd rswk-swf Analytische Zahlentheorie (DE-588)4001870-2 s DE-604 Graduate texts in mathematics 206 : Readings in mathematics (DE-604)BV000000067 206 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009321132&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Murty, Maruti Ram 1953- Problems in analytic number theory Graduate texts in mathematics Getaltheorie gtt Nombres, Théorie des - Problèmes et exercices Problemen gtt Number theory Analytische Zahlentheorie (DE-588)4001870-2 gnd |
| subject_GND | (DE-588)4001870-2 |
| title | Problems in analytic number theory |
| title_auth | Problems in analytic number theory |
| title_exact_search | Problems in analytic number theory |
| title_full | Problems in analytic number theory M. Ram Murty |
| title_fullStr | Problems in analytic number theory M. Ram Murty |
| title_full_unstemmed | Problems in analytic number theory M. Ram Murty |
| title_short | Problems in analytic number theory |
| title_sort | problems in analytic number theory |
| topic | Getaltheorie gtt Nombres, Théorie des - Problèmes et exercices Problemen gtt Number theory Analytische Zahlentheorie (DE-588)4001870-2 gnd |
| topic_facet | Getaltheorie Nombres, Théorie des - Problèmes et exercices Problemen Number theory Analytische Zahlentheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009321132&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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