Nathanson Melvyn B. Additive Number Theory: The Classical Bases

Springer Science & Business Media, 1996. — 342 p. — (Graduate Texts in Mathematics 164). — ISBN: 0-387-94656-X.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 learn, not for experts who already know it. For this reason, proofs include many "unnecessary" and "obvious" steps; this is by design.Notation and conventionsWaring's problem
Sums of polygons
Polygonal numbers
Lagrange's theorem
Quadratic forms
Ternary quadratic forms
Sums of three squares
Thin sets of squares
The polygonal number theorem
Notes
Exercises
Waring's problem for cubes
Sums of cubes
The Wieferich-Kempner theorem
Linnik's theorem
Sums of two cubes
Notes
Exercises
The Hilbert-Waring theorem
Polynomial identities and a conjecture of Hurwitz
Hermite polynomials and Hilbert's identity
A proof by induction
Notes
Exercises
Weyl's inequality
Tools
Difference operators
Easier Waring's problem
Fractional parts
Weyl's inequality and Hua's lemma
Notes
Exercises
The Hardy-Littlewood asymptotic formula
The circle method
Waring's problem for k = 1
The Hardy-Littlewood decomposition
The minor arcs
The major arcs
The singular integral
The singular seriesNotes
ExercisesThe Goldbach conjecture
Elementary estimates for primes
Euclid's theorem
Chebyshev's theorem
Mertens's theorems
Brun's method and twin primes
Notes
Exercises
The Shnirel'man-Goldbach theorem
The Goldbach conjecture
The Selberg sieve
Applications of the sieve
Shnirel'man density
The Shnirel'man-Goldbach theorem
Romanov's theorem
Covering congruences
Notes
Exercises
Sums of three primes
Vinogradov's theorem
The singular series
Decomposition into major and minor arcs
The integral over the major arcs
An exponential sum over primes
Proof of the asymptotic formula
Notes
Exercise
The linear sieve
A general sieve
Construction of a combinatorial sieve
Approximations
The Jurkat-Richert theorem
Differential-difference equations
Notes
Exercises
Chen's theorem
Primes and almost primes
Weights
Prolegomena to sieving
A lower bound for S(A, P, z)
An upper bound for S(A_q, P, z)
An upper bound for S(B, P, y)
A bilinear form inequalityNotes
Appendix
Arithmetic functions
The ring of arithmetic functions
Sums and integrals
Multiplicative functions
The divisor function
The Euler rp-function
The Mobius function
Ramanujan sums
Infinite products
Notes
ExercisesBibliography
Index