Additive number theory: [1] The classical bases
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York [u.a.]
Springer
1996
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| Series: | Graduate texts in mathematics
164 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIV, 342 S. |
| ISBN: | 038794656X |
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Record in the Search Index
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| adam_text | Contents
Preface vii
Notation and conventions xiii
I Waring s problem
1 Sums of polygons 3
1.1 Polygonal numbers 4
1.2 Lagrange s theorem 5
1.3 Quadratic forms 7
1.4 Ternary quadratic forms 12
1.5 Sums of three squares 17
1.6 Thin sets of squares 24
1.7 The polygonal number theorem 27
1.8 Notes 33
1.9 Exercises 34
2 Waring s problem for cubes 37
2.1 Sums of cubes 37
2.2 The Wieferich-Kempner theorem 38
2.3 Linnik s theorem 44
2.4 Sums of two cubes 49
2.5 Notes 71
2.6 Exercises 72
3 The Hilbert-Waring theorem 75
3.1 Polynomial identities and a conjecture of Hurwitz 75
3.2 Hermite polynomials and Hilbert s identity 77
3.3 A proof by induction 86
3.4 Notes 94
x Contents
3.5 Exercises 94
4 Weyl s inequality 97
4.1 Tools 97
4.2 Difference operators 99
4.3 Easier Waring s problem 102
4.4 Fractional parts 103
4.5 Weyl s inequality and Hua s lemma Ill
4.6 Notes 118
4.7 Exercises 118
5 The Hardy-Littlewood asymptotic formula 121
5.1 The circle method 121
5.2 Waring s problem for k = 1 124
5.3 The Hardy-Littlewood decomposition 125
5.4 The minor arcs 127
5.5 The major arcs 129
5.6 The singular integral 133
5.7 The singular series 137
5.8 Conclusion 146
5.9 Notes 147
5.10 Exercises 147
II The Goldbach conjecture
6 Elementary estimates for primes 151
6.1 Euclid s theorem 151
6.2 Chebyshev s theorem 153
6.3 Mertens s theorems 158
6.4 Brun s method and twin primes 167
6.5 Notes 173
6.6 Exercises 174
7 The Shnirel man-Goldbach theorem 177
7.1 The Goldbach conjecture 177
7.2 The Selberg sieve 178
7.3 Applications of the sieve 186
7.4 Shnirel man density 191
7.5 The Shnirel man-Goldbach theorem 195
7.6 Romanov s theorem 199
7.7 Covering congruences 204
7.8 Notes 208
7.9 Exercises 208
Contents xi
8 Sums of three primes 211
8.1 Vinogradov s theorem 211
8.2 The singular series 212
8.3 Decomposition into major and minor arcs 213
8.4 The integral over the major arcs 215
8.5 An exponential sum over primes 220
8.6 Proof of the asymptotic formula 227
8.7 Notes 230
8.8 Exercise 230
9 The linear sieve 231
9.1 A general sieve 231
9.2 Construction of a combinatorial sieve 238
9.3 Approximations 244
9.4 The Jurkat-Richert theorem 251
9.5 Differential-difference equations 259
9.6 Notes 267
9.7 Exercises 267
10 Chen s theorem 271
10.1 Primes and almost primes 271
10.2 Weights 272
10.3 Prolegomena to sieving 275
10.4 A lower bound for S(A,V,z) 279
10.5 An upper bound for S(Aq, V,z) 281
10.6 An upper bound for S(B,V,y) 286
10.7 A bilinear form inequality 292
10.8 Conclusion 297
10.9 Notes 298
III Appendix
Arithmetic functions 301
A.I The ring of arithmetic functions 301
A.2 Sums and integrals 303
A.3 Multiplicative functions 308
A.4 The divisor function 310
A.5 The Euler ^-function 314
A.6 The Mobius function 317
A.7 Ramanujan sums 320
A.8 Infinite products 323
A.9 Notes 327
A.10 Exercises 327
xii Contents
Bibliography 331
Index 341
|
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| id | DE-604.BV023854666 |
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| indexdate | 2024-07-09T21:38:16Z |
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| isbn | 038794656X |
| language | English |
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| publishDate | 1996 |
| publishDateSearch | 1996 |
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| publisher | Springer |
| record_format | marc |
| series | Graduate texts in mathematics |
| series2 | Graduate texts in mathematics |
| spelling | Nathanson, Melvyn B. Verfasser aut Additive number theory [1] The classical bases Melvyn B. Nathanson New York [u.a.] Springer 1996 XIV, 342 S. txt rdacontent n rdamedia nc rdacarrier Graduate texts in mathematics 164 Graduate texts in mathematics ... Additive Zahlentheorie (DE-588)4141387-8 gnd rswk-swf Additive Zahlentheorie (DE-588)4141387-8 s DE-604 (DE-604)BV023814363 1 Graduate texts in mathematics 164 (DE-604)BV000000067 164 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017518513&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Nathanson, Melvyn B. Additive number theory Graduate texts in mathematics Additive Zahlentheorie (DE-588)4141387-8 gnd |
| subject_GND | (DE-588)4141387-8 |
| title | Additive number theory |
| title_auth | Additive number theory |
| title_exact_search | Additive number theory |
| title_full | Additive number theory [1] The classical bases Melvyn B. Nathanson |
| title_fullStr | Additive number theory [1] The classical bases Melvyn B. Nathanson |
| title_full_unstemmed | Additive number theory [1] The classical bases Melvyn B. Nathanson |
| title_short | Additive number theory |
| title_sort | additive number theory the classical bases |
| topic | Additive Zahlentheorie (DE-588)4141387-8 gnd |
| topic_facet | Additive Zahlentheorie |
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