¼ÓÐÔÊýÂÛ-¾­µä»ù

preface
notation and conventior
i waring's problem
1 sums of polygor
1.1 polygonal number
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 operator
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 merter'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 applicatior of the sieve
7.4 shnirel'man derity
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 cortruction of a combinatorial sieve
9.3 approximatior
9.4 the jurkat-richert theorem
9.5 differential-difference equatior
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 functior
a.1 the ring of arithmetic functior
a.2 sums and integrals
a.3 multiplicative functior
a.4 the divisor function
a.5 the euler ¦Õ-function
a.6 the mobius function
a.7 ramanujan sums
a.8 infinite products
a.9 notes
a.10 exercises
bibliography
index