An Introduction to the Circle Method

Gadiyar and the reviewers for feedback on an earlier draft of this book. We are grateful to Arijit Chakraborty, Sneha Chaubey, Neha Prabhu and Sudhir Pujahari for their encouraging comments. This text grew out of winter schools and semester-long courses at IISER Kolkata, IISER Pune and Queen's University. We thank the students who participated in these courses. We thank Ina Mette and the American Mathematical Society for their interest in this book, and are grateful to Marcia Almeida, Christine Thivierge, and Abigail Lawson for help and correspondence related to the preparation of the manuscript. We thank John F. Brady for the beautiful cover design of the book. The second named author (K. S.) would like to acknowledge support from the MATRICS grant of the Science and Engineering Research Board, Department of Science and Technology, Government of India. xi This book is suitable for a one-semester undergraduate course. Students familiar with elementary number theory can read Chapter 1 for Preface xv an overview of the contents of the book, and jump directly to Chapter 5 where we derive some classical theorems of analytic number theory used later in the book. Students who have not seen any number theory before can read Chapters 2-4 for a quick introduction to topics in elementary number theory that will be used later in the textbook. Chapters 6-10 formed the bulk of our short courses. Chapter 6 contains a solution of Waring's problem using some ideas of Joseph H. Schnirelmann and Yuri V. Linnik. Chapters 7-10 describe the application of the circle method to Waring's problem and the ternary Goldbach conjecture. Finally, in Chapter 11, we provide the reader a lightning view of the origins of the circle method and describe the underlying philosophy of the method in a general way. We also indicate future directions and avenues of further study, along with several references for the student to explore this topic in greater depth. This chapter is aimed at the advanced student who wants to have a panoramic understanding of the method after having studied the more technical aspects treated in Chapters 6 and beyond. This chapter can also be read by the non-expert to gain a cursory understanding of the method without too many technicalities. Contents Preface Index of notations Chapter 1. Introduction and overview 1.1. Introduction 1.2. Preparatory chapters 1.3. Early developments in the study of Waring’s problem 1.4. The method of exponential sums 1.5. Origins of the circle method and applications to additive problems Chapter 2. Fundamental theorem of arithmetic 2.1. Mathematical induction 2.2. Divisibility 2.3. Greatest common divisor 2.4. Prime numbers and unique factorization Chapter 3. Arithmetic functions 3.1. Multiplicative functions 3.2. Möbius function and Möbius inversion 3.3. Greatest integer function 3.4. The big-Ox and little-ox notations 3.5. Averages of arithmetical functions 3.6. Technique of partial summation 3.7. The Cauchy–Schwarz and Hölder inequalities Chapter 4. Introduction to congruence arithmetic 4.1. Definition and basic properties of congruences 4.2. Congruence powers and Euler’s theorem 4.3. Linear congruence equations 4.4. Linear congruences and the Chinese remainder theorem 4.5. Polynomial congruences 4.6. Order and primitive roots Chapter 5. Distribution of prime numbers 5.1. Dirichlet series 5.2. Euler products and Dirichlet series 5.3. Analytic properties of Dirichlet series 5.4. Distribution functions for prime numbers 5.5. Primes in arithmetic progressions 5.6. Dirichlet characters and Dirichlet L-functions 5.7. Ramanujan sums and Ramanujan series Chapter 6. An introduction to Waring’s problem 6.1. Fermat’s two square theorem 6.2. Lagrange’s four square theorem 6.3. A conjectured value for g(k) 6.4. The easier Waring’s problem Chapter 7. Waring’s problem 7.1. Schnirelmann density 7.2. Schnirelmann density and Waring’s problem 7.3. Proof of Linnik’s theorem Chapter 8. Exponential sums 8.1. Exponential sums for polynomials of degree 1 8.2. Exponential sums and Diophantine approximation 8.3. Exponential sums over primes Chapter 9. The circle method and Waring’s problem 9.1. An outline of the circle method 9.2. The contribution from the major arcs 9.3. The singular integral 9.4. Singular series 9.5. Minor arcs in Waring’s problem Chapter 10. The circle method and the Goldbach conjectures 10.1. Major and minor arcs 10.2. Contribution from the major arcs 10.3. Contribution from the minor arcs 10.4. Comments about Vinogradov’s theorem 10.5. The circle method and the binary Goldbach conjecture Chapter 11. Epilogue 11.1. The philosophy of the circle method 11.2. An axiomatic framework 11.3. The singular series 11.4. The minor arcs 11.5. The future of the circle method Bibliography Index Back cover