Ⅰ Waring's problem 3
1 Sums ofpolygons 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 ofthree squares 17
1.6 Thin sets of squares 24
1.7 The polygonal numbertheorem 27
1.8 Notes 33
1.9 Exercises 34
2 Waring'sproblem forcubes 37
2.1 Sums ofcubes 37
2.2 The Wieferich-Kempnertheorem 38
2.3 Linnik's theorem 44
2.4 Sums oftwo cubes 49
2.5 Notes 71
2.6 Exercises 72
3 The Hilbert-Waring theorem 75
3.1 Polynomial identities and a conjecture ofHurwitz 75
3.2 HermitepolynomialsandHilbert's identity 77
3.3 A proofby induction 86
3.4 Notes 94
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 111
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-l 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 151
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 Shnlrel?man-Goldbach theorem 177
7.1 The Goldbach conjecture 177
7.2 The Selberg sieve 178
7.3 Applications ofthe 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
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 Nores 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,P,z) 279
10.5 An upper bound for S(Aq,P,z) 281
10.6 An upper bound for S(B,P,y) 286
10.7 A bilinear form inequality 292
10.8 Conclusion 297
10.9 Notes 298
Ⅲ Appendix 301
Arithmetic funetions 301
A.1 The ring of arithtmetic 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 M？bius function 317
A.7 Ramanujan sums 320
A.8 Infinite products 323
A.9 Nores 327
A.10 Exercises 327
Bibliography 331
Index 341