Additive Number Theory The Classical Bases (Graduate Texts in Mathematics (164))

[Hilbert's] style has not the terseness of many of our modem authors in mathematics, which is based on the assumption that printer's labor and paper are costly but the reader's effort and time are not. H. Weyl [143] 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 used to attack these problems. This book is intended for students who want to lel?Ill additive number theory, not for experts who already know it. For this reason, proofs include many "unnecessary" and "obvious" steps; this is by design. The archetypical theorem in additive number theory is due to Lagrange: Every nonnegative integer is the sum of four squares. In general, the set A of nonnegative integers is called an additive basis of order h if every nonnegative integer can be written as the sum of h not necessarily distinct elements of A. Lagrange 's theorem is the statement that the squares are a basis of order four. The set A is called a basis offinite order if A is a basis of order h for some positive integer h. Additive number theory is in large part the study of bases of finite order. The classical bases are the squares, cubes, and higher powers; the polygonal numbers; and the prime numbers. The classical questions associated with these bases are Waring's problem and the Goldbach conjecture. Cover......Page 1 Title Page......Page 4 Copyright Page......Page 5 Dedication......Page 6 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 Back Cover......Page 358 Many classical problems in additive number theory are direct problems, in which one starts with a set A of natural numbers and an integer h[actual symbol not reproducible]2 and tries to describe the structure of the sumset hA consisting of all sums of h elements of A. In contrast, in an inverse problem, one starts with a sumset hA and attempts to describe the structure of the underlying set A. In recent years, there has been remarkable progress in the study of inverse problems for finite sets of integers. In particular, there are important and beautiful inverse theorems due to Freiman, Kneser, Plunnecke, Vospel and others. This volume includes their results and culminates with an elegant proof by Rusza of the deep theorem of Freiman that a finite set of integers with a small sumset must be a large subset of an n-dimensional arithmetic progression. . Inverse problems are a central topic in additive number theory. This graduate text gives a comprehensive and self-contained account of this subject. In particular, it contains complete proofs of results from exterior algebra, combinatorics, graph theory, and the geometry of numbers that are used in the proofs of the principal inverse theorems. The only prerequisites for the book are undergraduate courses in algebra, number theory, and analysis. "The classical bases in additive number theory are the polygonal numbers, the squares, cubes, and higher powers, and the primes. This book contains many of the great theorems in this subject: Cauchy's polygonal number theorem, Linnik's theorem on sums of cubes, Hilbert's proof of Waring's problem, the Hardy-Littlewood asymptotic formula for the number of representations of an integer as the sum of positive kth powers, Shnirel'man's theorem that every integer greater than one is the sum of a bounded number of primes, Vinogradov's theorem on sums of three primes, and Chen's theorem that every sufficiently large even integer is the sum of a prime and a number that is either prime or the product of two primes. The book is also an introduction to the circle method and sieve methods, which are the principal tools used to study the classical bases. The only prerequisites for the book are undergraduate courses in number theory and analysis. Additive number theory is one of the oldest and richest areas of mathematics. This book is the first comprehensive treatment of the subject in 40 years."--Page 4 de la couverture

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.