加性數論：經典基（簡體書）

加性數論討論的是很經典的論題，本書討論了相關理論的最新進展和科研成果，並且用Freiman定理的Ruzsa證明將本書的內容推向了高潮。111111111111111111111111111111111111111111111111111

作者:(美)納森

《加性數論(經典基)》分為上下2卷。堆壘數論討論的是很經典的直接問題。在這個問題中，首先假定有一個自然數集合a和大于等于2的整數h，定義的和集ha是由所有的h和a中元素乘積的和組成，試圖描述和集ha的結構；相反地，在逆問題中，從和集ha開始，去尋找這樣的一個集合a。近年來，有關整數有限集的逆問題方面取得了顯著進展。特別地，freiman, kneser, plünnecke, vosper以及一些其他的學者在這方面做出了突出的貢獻。本書中包括了這些結果，并且用freiman定理的ruzsa證明將本書的內容推向了高潮。本書由納森著。

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