The purpose of this book is to describe the classical problems in additive number theory, and to introduce the circle method and the sieve method, which are the basic analytical and combinatorial tools to attack these problems. This book is intended for students who want to learn additive number theory, not for experts who already know it. The prerequisites for this book are undergraduate courses in number theory and real analysis.
Author(s): Melvyn B. Nathanson
Series: Graduate Texts in Mathematics
Edition: 1
Publisher: Springer
Year: 1996
Language: English
Pages: 358
Preface ......Page 8
Contents ......Page 10
Notation and conventions ......Page 14
I Waring's problem ......Page 16
1 Sums of polygons ......Page 18
1.1 Polygonal numbers ......Page 19
1.2 Lagrange's theorem ......Page 20
1.3 Quadratic forms ......Page 22
1.4 Ternary quadratic forms ......Page 27
1.5 Sums of three squares ......Page 32
1.6 Thin sets of squares ......Page 39
1.7 The polygonal number theorem ......Page 42
1.8 Notes ......Page 48
1.9 Exercises ......Page 49
2.1 Sums of cubes ......Page 52
2.2 The Wieferich-Kempner theorem ......Page 53
2.3 Linnik's theorem ......Page 59
2.4 Sums of two cubes ......Page 64
2.5 Notes ......Page 86
2.6 Exercises ......Page 87
3.1 Polynomial identities and a conjecture of Hurwitz ......Page 90
3.2 Hermite polynomials and Hilbert's identity ......Page 92
3.3 A proof by induction ......Page 101
3.5 Exercises ......Page 109
4.1 Tools ......Page 112
4.2 Difference operators ......Page 114
4.3 Easier Waring's problem ......Page 117
4.4 Fractional parts ......Page 118
4.5 Weyl's inequality and Hua's lemma ......Page 126
4.7 Exercises ......Page 133
5.1 The circle method ......Page 136
5.2 Waring's problem for k II I ......Page 139
5.3 The Hardy-Littlewood decomposition ......Page 140
5.4 The minor arcs ......Page 142
5.5 The major arcs ......Page 144
5.6 The singular integral ......Page 148
5.7 The singular series ......Page 152
5.8 Conclusion ......Page 161
5.10 Exercises ......Page 162
II The Goldbach conjecture ......Page 164
6.1 Euclid's theorem ......Page 166
6.2 Chebyshev's theorem ......Page 168
6.3 Mertens's theorems ......Page 173
6.4 Brun's method and twin primes ......Page 182
6.5 Notes ......Page 188
6.6 Exercises ......Page 189
7.1 The Goldbach conjecture ......Page 192
7.2 The Selberg sieve ......Page 193
7.3 Applications of the sieve ......Page 201
7.4 Shnirel'man density ......Page 206
7.5 The Shnirel'man-Goldbach theorem ......Page 210
7.6 Romanov's theorem ......Page 214
7.7 Covering congruences ......Page 219
7.9 Exercises ......Page 223
8.1 Vinogradov's theorem ......Page 226
8.2 The singular series ......Page 227
8.3 Decomposition into major and minor arcs ......Page 228
8.4 The integral over the major arcs ......Page 230
8.5 An exponential sum over primes ......Page 235
8.6 Proof of the asymptotic formula ......Page 242
8.8 Exercise ......Page 245
9.1 A general sieve ......Page 246
9.2 Construction of a combinatorial sieve ......Page 253
9.3 Approximations ......Page 259
9.4 The Jurkat-Richert theorem ......Page 266
9.5 Differential-difference equations ......Page 274
9.7 Exercises ......Page 282
10.1 Primes and almost primes ......Page 286
10.2 Weights ......Page 287
10.3 Prolegomena to sieving ......Page 290
10.4 A lower bound for S(A, P, z) ......Page 294
10.5 An upper bound for S(Aq, P, z) ......Page 296
10.6 An upper bound for S(B, P, y) ......Page 301
10.7 A bilinear form inequality ......Page 307
10.8 Conclusion ......Page 312
10.9 Notes ......Page 313
III Appendix......Page 314
A.1 The ring of arithmetic functions ......Page 316
A.2 Sums and integrals ......Page 318
A.3 Multiplicative functions ......Page 323
A.4 The divisor function ......Page 325
A.5 The Euler rp-function ......Page 329
A.6 The Mobius function ......Page 332
A.7 Ramanujan sums ......Page 335
A.8 Infinite products ......Page 338
A.10 Exercises ......Page 342
Bibliography ......Page 346
Index ......Page 356
