# Book:Melvyn B. Nathanson/Additive Number Theory: The Classical Bases

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## Melvyn B. Nathanson:

## Melvyn B. Nathanson: *Additive Number Theory: The Classical Bases*

Published $\text {1996}$, **Springer-Verlag**

### Subject Matter

### Publisher's Description

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.

### Contents

- Preface
- Notation and conventions
- I. Waring’s problem
- 1. Sums of polygons
- 1. Polygonal numbers
- 2. Lagrange’s theorem
- 3. Quadratic forms
- 4. Ternary quadratic forms
- 5. Sums of three squares
- 6. Thin sets of squares
- 7. The polygonal number theorem
- 8. Notes
- 9. Exercises

- 2. Waring’s problem for cubes
- 1. Sums of cubes
- 2. The Wieferich-Kempner theorem
- 3. Linnik’s theorem
- 4. Sums of two cubes
- 5. Notes
- 6. Exercises

- 3. The Hilbert-Waring theorem
- 1. Polynomial identities and a conjecture of Hurwitz
- 2. Hermite polynomials and Hilbert’s identity
- 3. A proof by induction
- 4. Notes
- 5. Exercises

- 4. Weyl’s inequality
- 1. Tools
- 2. Difference operators
- 3. Easier Waring’s problem
- 4. Fractional parts
- 5. Weyl’s inequality and Hua’s lemma
- 6. Notes
- 7. Exercises

- 5. The Hardy-Littlewood asymptotic formula
- 1. The circle method
- 2. Waring’s problem for $k = 1$
- 3. The Hardy-Littlewood decomposition
- 4. The minor arcs
- 5. The major arcs
- 6. The singular integral
- 7. The singular series
- 8. Conclusion
- 9. Notes
- 10. Exercises

- 1. Sums of polygons
- II. The Goldbach conjecture
- 1. Elementary estimates for primes
- 1. Euclid’s theorem
- 2. Chebyshev’s theorem
- 3. Mertens’s theorems
- 4. Brun’s method and twin primes
- 5. Notes
- 6. Exercises

- 2. The Shnirel’man-Goldbach theorem
- 1. The Goldbach conjecture
- 2. The Selberg sieve
- 3. Applications of the sieve
- 4. Shnirel’man density
- 5. The Shnirel’man-Goldbach theorem
- 6. Romanov’s theorem
- 7. Covering congruences
- 8. Notes
- 9. Exercises

- 3. Sums of three primes
- 1. Vinogradov’s theorem
- 2. The singular series
- 3. Decomposition into major and minor arcs
- 4. The integral over the major arcs
- 5. An exponential sum over primes
- 6. Proof of the asymptotic formula
- 7. Notes
- 8. Exercise

- 4. The linear sieve
- 1. A general sieve
- 2. Construction of a combinatorial sieve
- 3. Approximations
- 4. The Jurkat-Richert theorem
- 5. Differential-difference equations
- 6. Notes
- 7. Exercises

- 5. Chen’s theorem
- 1. Primes and almost primes
- 2. Weights
- 3. Prolegomena to sieving
- 4. A lower bound for $S(A, \mathcal P, z)$
- 5. An upper bound for $S(A_q, \mathcal P, z)$
- 6. An upper bound for $S(B, \mathcal P, y)$
- 7. A bilinear form inequality
- 8. Conclusion
- 9. Notes

- 1. Elementary estimates for primes
- III. Appendix: Arithmetic functions
- 1. The ring of arithmetic functions
- 2. Sums and integrals
- 3. Multiplicative functions
- 4. The divisor function
- 5. The Euler $\phi$-function
- 6. The Möbius function
- 7. Ramanujan sums
- 8. Infinite products
- 9. Notes
- 10. Exercises

- Bibliography
- Index