Analytic Number Theory for Beginners: Second Edition

This new edition of Analytic Number Theory for Beginners presents a friendly introduction to analytic number theory for both advanced undergraduate and beginning graduate students, and offers a comfortable transition between the two levels. The text starts with a review of elementary number theory and continues on to present less commonly covered topics such as multiplicative functions, the floor function, the use of big $O$, little $o$, and Vinogradov notation, as well as summation formulas. Standard advanced topics follow, such as the Dirichlet $L$-function, Dirichlet's Theorem for primes in arithmetic progressions, the Riemann Zeta function, the Prime Number Theorem, and, new in this second edition, sieve methods and additive number theory. The book is self-contained and easy to follow. Each chapter provides examples and exercises of varying difficulty and ends with a section of notes which include a chapter summary, open questions, historical background, and resources for further study. Since many topics in this book are not typically covered at such an accessible level, Analytic Number Theory for Beginners is likely to fill an important niche in today's selection of titles in this field. Cover 1 Title page 4 Copyright 5 Contents 8 Preface to the Second Edition and Acknowledgment 12 Preface to the First Edition 14 Acknowledgments 18 Notation 20 Chapter 1. Review of Elementary Number Theory 26 1.1. Divisibility 26 1.2. Greatest Common Divisor 27 1.3. Least Common Multiple 31 1.4. Prime Numbers 31 1.5. Congruences 35 1.6. Residue Systems 37 1.7. Chinese Remainder Theorem 38 1.8. Polynomial and Some Special Congruences 42 1.9. Lifting the Exponents 45 1.10. Primitive Roots 47 1.11. Quadratic Residues 48 1.12. Sum of Squares 49 1.13. Exercises 50 1.14. Notes 51 Chapter 2. Arithmetic Functions I 58 2.1. Introduction 58 2.2. Multiplicative Functions 61 2.3. Dirichlet Product 63 2.4. Divisor Sums and the Möbius Inversion Formula 67 2.5. Extensions of the Möbius Inversion Formula 74 2.6. Exercises 77 2.7. Notes 81 Chapter 3. The Floor Function 90 3.1. Introduction 90 3.2. Hermite’s Identity and Generalizations 96 3.3. Counting Lattice Points 104 3.4. Miscellaneous Examples 110 3.5. Exercises 115 3.6. Notes 118 Chapter 4. Summation Formulas 124 4.1. Introduction 124 4.2. Big O, Little o, and Related Notations 125 4.3. Sums of Monotone Functions 129 4.4. Partial Summation Formula 133 4.5. Review of the Riemann-Stieltjes Integral 137 4.6. Euler Summation Formula 139 4.7. Euler-Maclaurin Summation Formula 142 4.8. Exercises 146 4.9. Notes 148 Chapter 5. Arithmetic Functions II 154 5.1. Introduction 154 5.2. Average Orders 156 5.3. Dirichlet’s Hyperbola Method 162 5.4. Extremal Orders 166 5.5. Normal Orders 175 5.6. Exercises 179 5.7. Notes 186 Chapter 6. Elementary Results on the Distribution of Primes 194 6.1. Introduction 194 6.2. Chebyshev’s Estimates 198 6.3. Mertens’ Theorems 203 6.4. Exercises 206 6.5. Notes 209 Chapter 7. Characters and Dirichlet’s Theorem 212 7.1. Introduction 212 7.2. The Orthogonality Relations 217 7.3. Dirichlet Characters 219 7.4. Dirichlet L-Functions 223 7.5. Dirichlet’s Theorem 227 7.6. Applications of Dirichlet’s Theorem 233 7.7. The Statement of the Green-Tao Theorem 239 7.8. Exercises 240 7.9. Notes 246 Chapter 8. The Riemann Zeta Function 254 8.1. Introduction 254 8.2. Half-Plane of Convergence 256 8.3. Review of Infinite Products 263 8.4. Euler Product Formula 267 8.5. Analytic Continuation 272 8.6. Zero Free Regions 274 8.7. Exercises 276 8.8. Notes 278 Chapter 9. Prime Number Theorem and Some Extensions 282 9.1. Introduction 282 9.2. Proof of the Prime Number Theorem 284 9.3. Elementary Proof of the Prime Number Theorem 289 9.4. Stronger Forms of the Prime Number Theorem 292 9.5. Integers Having k Prime Factors 294 9.6. Exercises 305 9.7. Notes on Prime in Arithmetic Progressions 310 9.8. Analogues of the Bombieri-Vinogradov Theorem 313 9.9. Notes on Bounded Gaps Between Primes 315 Chapter 10. Introduction to Other Topics 318 10.1. Idea of Sieves 318 10.2. Brun’s Sieve 320 10.3. Selberg’s Sieve 328 10.4. Introduction to Additive Number Theory 338 10.5. Schnirelmann’s Basis Theorem 346 10.6. An Outline of the Circle Method in Waring’s Problem 354 Chapter 11. Hints for Selected Exercises 360 Hints for Exercises 1.13 360 Hints for Exercises 2.6 361 Hints for Exercises 3.5 362 Hints for Exercises 4.8 364 Hints for Exercises 5.6 365 Hints for Exercises 6.4 367 Hints for Exercises 7.8 369 Hints for Exercises 8.7 371 Bibliography 372 Index 392 Index 396 Back Cover 402