نه همیشه عمیق دفن شده است: گزیده‌هایی از نظریه عددی تحلیلی و ترکیبی

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