Not always buried deep.. selections from analytic and combinatorial number theory

Primary References......Page 7
I Analytic Number Theory......Page 16
Introduction......Page 18
There are Infinitely Many Primes......Page 19
Euclid and his Imitators......Page 20
Coprime Integer Sequences......Page 21
The Riemann Zeta Function......Page 23
Squarefree Numbers......Page 28
Smooth Numbers......Page 31
The Heavy Machinery......Page 33
Exercises......Page 34
An Empirical Approach......Page 36
Exercises: Some Consequences of the PNT......Page 39
The Primes are Infinitely Fewer than the Integers......Page 40
More Primes than Squares......Page 42
Chebyshev's Work on pi(x)......Page 43
Proof of Theorem 1.5.1......Page 46
Proof of Theorem 1.5.2......Page 47
Proof of Bertrand's Postulate......Page 48
Exercises......Page 49
Polynomials that Represent Many Primes......Page 52
Mertens' Theorem, sans the Constant......Page 54
The Constant in Mertens' Theorem......Page 57
Exercises......Page 60
Primes in Arithmetic Progressions......Page 61
The Twin Prime and Goldbach Problems......Page 62
An Extended Hardy-Littlewood Conjecture......Page 64
Exercises: More on the Bateman-Horn Conjecture......Page 66
The Prime Ideal Theorem......Page 67
Chebyshev Analogs......Page 68
Exercises......Page 71
The Prime Number Theorem......Page 72
Exercises: Further Elementary Estimates......Page 74
New Irreducibles from Old......Page 76
The Twin Prime Problem......Page 78
Exercises: Proof of Theorem 1.10.13......Page 80
References......Page 82
Introduction and a Special Case......Page 88
The Case of Progressions mod 4......Page 89
Exercises......Page 91
The Classification of Characters......Page 93
The Orthogonality Relations......Page 94
Dirichlet Characters......Page 96
Exercises......Page 97
The L-series at s=1......Page 99
The Nonvanishing of L(1,chi) for complex......Page 100
The Nonvanishing of L(1,chi) for real, nonprincipal......Page 103
Exercises......Page 106
Sums of Three Squares......Page 108
Quadratic Forms......Page 109
Equivalent Forms......Page 110
Bilinear Forms on Z^n......Page 111
Forms of Determinant 1......Page 113
Proof of the Three Squares Theorem......Page 116
Completion of The Proof......Page 118
The Number of Representations......Page 120
Exercises......Page 121
References......Page 122
Legendre's Formula......Page 126
Consequences......Page 128
General Sieving Situations......Page 129
Legendre, Brun and Hooley; oh my!......Page 130
Further Reading......Page 131
Exercises......Page 132
The General Sieve Problem......Page 133
A First Sieve Result......Page 134
Three Number-Theoretic Applications......Page 136
Exercises......Page 139
Preparation......Page 140
A Working Version......Page 143
Application to the Twin Prime Problem (outline)......Page 144
Proof of Theorem 3.4.8......Page 145
Exercises......Page 148
The Sifting Function Perspective......Page 149
The Upper Bound......Page 150
Applications of the Upper Bound......Page 152
The Lower Bound......Page 158
Applications of the Lower Bound......Page 160
Exercises: Further Applications of the Brun-Hooley Sieve......Page 164
References......Page 166
Introduction......Page 168
Exercises......Page 170
Equivalent Forms of the Prime Number Theorem......Page 172
An Inversion Formula and its Consequences......Page 173
An Estimate of Dirichlet......Page 174
Proof of the Equivalences......Page 175
Exercises......Page 179
An Upper Bound on pi(x+y) - pi(x)......Page 180
Preparatory Lemmas......Page 181
Proof of Lemma 4.3.1 by Selberg's sieve......Page 183
A General Version of Selberg's Sieve (optional)......Page 187
The Turan-Kubilius Inequality......Page 189
Exercises: The Orders of nu and Omega (optional)......Page 191
Preliminary Lemmas......Page 193
The Fundamental Lemma......Page 196
Preparation......Page 197
Construction of P, P'......Page 198
Estimation of S, S'......Page 200
Estimation of S M(x) - S' M(x')......Page 202
Denouement......Page 203
References......Page 204
II Additive and Combinatorial Number Theory......Page 206
Introduction......Page 208
Exercises......Page 210
The Polynomial Method of Alon, Nathanson, Rusza......Page 211
Chowla's Sumset Addition Theorem......Page 214
Waring's Problem for Residues......Page 215
Exercises: More on Waring's Problem for Residues......Page 216
Schnirelmann Density and Additive Bases......Page 217
Mann's Density Theorem......Page 219
Asymptotic Bases......Page 223
Exercises......Page 224
A Special Class of Additive Bases......Page 225
Schnirelmann's Contribution to Goldbach's Problem......Page 227
Romanov's Theorem......Page 230
The Theorems of Erdos and Crocker......Page 233
A Lemma in Graph Theory......Page 235
Combinatorial Consequences......Page 237
Application to the Fermat Congruence......Page 238
References......Page 239
Introduction......Page 244
Exercises......Page 246
Equivalent Forms of van der Waerden's Theorem......Page 247
A Proof of van der Waerden's Theorem......Page 249
Exercises......Page 253
Roth's Theorem and Affine Properties......Page 254
A Combinatorial Lemma......Page 257
Some Definitions......Page 258
Properties of the L_i......Page 261
Blocks and Gaps......Page 263
Denouement......Page 264
The Behavior of the Extremal Sets......Page 265
Roth's Theorem revisited......Page 267
The Function e(theta)......Page 268
Parseval's Formula......Page 269
Exercises......Page 270
Further Preliminaries......Page 272
Proof of The Fundamental Lemma......Page 274
Proof of Lemma 6.5.3......Page 277
The Number of Three Term Progressions......Page 279
The Higher-Dimensional Situation......Page 281
Exercises......Page 284
Behrend's Lower Bound for r_3(n)......Page 285
References......Page 288
Introduction......Page 292
Exercises......Page 294
The Linnik-Newman Approach......Page 295
A Simplified Estimate of the Weyl Sums......Page 297
Completion of the Proof......Page 300
Exercises......Page 303
References......Page 304
III Appendices......Page 306
Big-Oh notation......Page 308
Little-oh notation......Page 309
Comparison of a Sum and an Integral......Page 310
Partial Summation......Page 311
in Homothetically Expanding Regions......Page 313
enclosed by a Jordan curve......Page 314
References......Page 315
The Fundamental Theorem......Page 316
Free Z-modules of Finite Rank......Page 318