The development of prime number theory: from Euclid to Hardy and Littlewood
"This book presents the development of Prime Number Theory from its beginnings until the end of the first decade of the xxth century. Special emphasis is given to the work of Cebysev, Dirichlet, Riemann, Vallee-Poussin, Hadamard and Landau. The book presents the principal results with proofs an...
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Berlin [u.a.]
Springer
2000
|
| Schriftenreihe: | Springer Monographs in mathematics
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Zusammenfassung: | "This book presents the development of Prime Number Theory from its beginnings until the end of the first decade of the xxth century. Special emphasis is given to the work of Cebysev, Dirichlet, Riemann, Vallee-Poussin, Hadamard and Landau. The book presents the principal results with proofs and also gives, mostly in short comments, an overview of the development in the last 80 years. It is, however, not a historical book since it does not give biographical details of the people who have played a role in the development of Prime Number Theory. The book contains a large list of references with more than 1800 items. It can be read by any person with a knowledge of fundamental notions of number theory and complex analysis."--BOOK JACKET. |
| Beschreibung: | XII, 448 S. |
| ISBN: | 3540662898 |
Internformat
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Datensatz im Suchindex
| _version_ | 1804127776914014208 |
|---|---|
| adam_text | Table of Contents
Preface v
Table of Contents ix
Notation xi
1. Early Times 1
1.1. The Infinitude of Prime Numbers 1
1.2. Sum of Reciprocals of Primes 11
1.3. Primitive Roots 15
1.4. Prime Number Formulas 23
Exercises 44
2. Dirichlet s Theorem on Primes
in Arithmetic Progressions 49
2.1. Progressions with a Prime Difference 49
2.2. The Non vanishing of Lj(l) in Case of a Prime Difference .. 61
2.3. The Case of an Arbitrary Difference 66
2.4. Elementary Proofs of L(l, x) / 0 76
2.5. Elementary Approaches for Particular Moduli 87
Exercises 93
3. Cebysev s Theorem 97
3.1. The Conjecture of Legendre 97
3.2. True Order of n(x) 103
3.3. Applications of Cebysev s Theorem 124
Exercises 130
4. Riemann s Zeta function and Dirichlet Series 133
4.1. The Zeta function; Riemann s Memoir 133
4.2. Dirichlet s L functions in the Complex Plane 148
4.3. Stieltjes, Cahen, Phragmen 157
Exercises 178
X Table of Contents
5. The Prime Number Theorem 183
5.1. Hadamard s First Paper on the Zeta function
and its Consequences 183
5.2. von Mangoldt 188
5.3. Hadamard s Proof 198
5.4. The Proof of de la Vallee Poussin 207
5.5. Other Proofs of the Non vanishing of £(1 + it)
and L(l+it,x) 219
5.6. Bounding the Error Term 230
Exercises 252
6. The Turn of the Century 257
6.1 Progress in Function Theory 257
6.2. Landau s Approach to the Prime Number Theorem 272
6.3. von Mangoldt s Theorems Revisited 287
6.4. Tauberian Methods 298
6.5. Zeros of the Zeta function 314
6.6. The Sign of n(x) (x) 322
6.7 The Conjectures of Hardy and Littlewood 333
Exercises 350
References 355
Author Index 435
Subject Index 445
|
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| indexdate | 2024-07-09T18:38:44Z |
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| isbn | 3540662898 |
| language | English |
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| spelling | Narkiewicz, Władysław 1936- Verfasser (DE-588)11036676X aut The development of prime number theory from Euclid to Hardy and Littlewood Władysław Narkiewicz Berlin [u.a.] Springer 2000 XII, 448 S. txt rdacontent n rdamedia nc rdacarrier Springer Monographs in mathematics "This book presents the development of Prime Number Theory from its beginnings until the end of the first decade of the xxth century. Special emphasis is given to the work of Cebysev, Dirichlet, Riemann, Vallee-Poussin, Hadamard and Landau. The book presents the principal results with proofs and also gives, mostly in short comments, an overview of the development in the last 80 years. It is, however, not a historical book since it does not give biographical details of the people who have played a role in the development of Prime Number Theory. The book contains a large list of references with more than 1800 items. It can be read by any person with a knowledge of fundamental notions of number theory and complex analysis."--BOOK JACKET. Getaltheorie gtt Nombres premiers Priemgetallen gtt Verdelingen (functietheorie) gtt Numbers, Prime Primzahltheorie (DE-588)4175715-4 gnd rswk-swf Primzahltheorie (DE-588)4175715-4 s DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008910816&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Narkiewicz, Władysław 1936- The development of prime number theory from Euclid to Hardy and Littlewood Getaltheorie gtt Nombres premiers Priemgetallen gtt Verdelingen (functietheorie) gtt Numbers, Prime Primzahltheorie (DE-588)4175715-4 gnd |
| subject_GND | (DE-588)4175715-4 |
| title | The development of prime number theory from Euclid to Hardy and Littlewood |
| title_auth | The development of prime number theory from Euclid to Hardy and Littlewood |
| title_exact_search | The development of prime number theory from Euclid to Hardy and Littlewood |
| title_full | The development of prime number theory from Euclid to Hardy and Littlewood Władysław Narkiewicz |
| title_fullStr | The development of prime number theory from Euclid to Hardy and Littlewood Władysław Narkiewicz |
| title_full_unstemmed | The development of prime number theory from Euclid to Hardy and Littlewood Władysław Narkiewicz |
| title_short | The development of prime number theory |
| title_sort | the development of prime number theory from euclid to hardy and littlewood |
| title_sub | from Euclid to Hardy and Littlewood |
| topic | Getaltheorie gtt Nombres premiers Priemgetallen gtt Verdelingen (functietheorie) gtt Numbers, Prime Primzahltheorie (DE-588)4175715-4 gnd |
| topic_facet | Getaltheorie Nombres premiers Priemgetallen Verdelingen (functietheorie) Numbers, Prime Primzahltheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008910816&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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