《加性数论(经典基)》分为上下2卷。堆垒数论讨论的是很经典的直接问题。在这个问题中，首先假定有一个自然数集合a和大于等于2的整数h，定义的和集ha是由所有的h和a中元素乘积的和组成，试图描述和集ha的结构；相反地，在逆问题中，从和集ha开始，去寻找这样的一个集合a。近年来，有关整数有限集的逆问题方面取得了显著进展。特别地，freiman, kneser, plünnecke, vosper以及一些其他的学者在这方面做出了突出的贡献。本书中包括了这些结果，并且用freiman定理的ruzsa证明将本书的内容推向了高潮。本书由纳森著。

preface

notation and conventions

i waring's problem

1 sums of polygons

1.1 polygonal numbers

1.2 lagrange's theorem

1.3 quadratic forms

1.4 ternary quadratic forms

1.5 sums of three squares

1.6 thin sets of squares

1.7 the polygonal number theorem

1.8 notes

1.9 exercises

2 waring's problem for cubes

2.1 sums of cubes

2.2 the wieferich-kempner theorem

2.3 linnik's theorem

2.4 sums of two cubes

2.5 notes

.2.6 exercises

3 the hilbert-waring theorem

3.1 polynomial identities and a conjecture of hurwitz

3.2 hermite polynomials and hilbert's identity

3.3 a proof by induction

3.4 notes

3.5 exercises

4 weyl's inequality

4.1 tools

4.2 difference operators

4.3 easier waring's problem

4.4 fractional parts

4.5 weyl's inequality and hua's lemma

4.6 notes

4.7 exercises

5 the hardy-littlewood asymptotic formula

5.1 the circle method

5.2 waring's problem for k = 1

5.3 the hardy-littlewood decomposition

5.4 the minor arcs

5.5 the major arcs

5.6 the singular integral

5.7 the singular series

5.8 conclusion

5.9 notes

5.10 exercises

ii the goldbach conjecture

6 elementary estimates for primes

6.1 euclid's theorem

6.2 chebyshev's theorem

6.3 mertens's theorems

6.4 brun's method and twin primes

6.5 notes

6.6 exercises

7 the shnirel'man-goldbach theorem

7.1 the goldbach conjecture

7.2 the selberg sieve

7.3 applications of the sieve

7.4 shnirel'man density

7.5 the shnirel'man-goldbach theorem

7.6 romanov's theorem

7.7 covering congruences

7.8 notes

7.9 exercises

8 sums of three primes

8.1 vinogradov's theorem

8.2 the singular series

8.3 decomposition into major and minor arcs

8.4 the integral over the major arcs

8.5 an exponential sum over primes

8.6 proof of the asymptotic formula

8.7 notes

8.8 exercise

9 the linear sieve

9.1 a general sieve

9.2 construction of a combinatorial sieve

9.3 approximations

9.4 the jurkat-richert theorem

9.5 differential-difference equations

9.6 notes

9.7 exercises

10 chen's theorem

10.1 primes and almost primes

10.2 weights

10.3 prolegomena to sieving

10.4 a lower bound for s(a, p, z)

10.5 an upper bound for s(aq, p, z)

10.6 an upper bound for s(b, p, y)

10.7 a bilinear form inequality

10.8 conclusion

10.9 notes

iii appendix

arithmetic functions

a.1 the ring of arithmetic functions

a.2 sums and integrals

a.3 multiplicative functions

a.4 the divisor function

a.5 the euler φ-function

a.6 the mobius function

a.7 ramanujan sums

a.8 infinite products

a.9 notes

a.10 exercises

bibliography

index