加性數(shù)論

內(nèi)容概要

　　《加性數(shù)論（經(jīng)典基）》分為上下2卷。堆壘數(shù)論討論的是很經(jīng)典的直接問題。在這個(gè)問題中，首先假定有一個(gè)自然數(shù)集合a和大于等于2的整數(shù)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(jié)構(gòu)；相反地，在逆問題中，從和集ha開始，去尋找這樣的一個(gè)集合a。近年來，有關(guān)整數(shù)有限集的逆問題方面取得了顯著進(jìn)展。特別地，freiman, kneser, pl＆uuml;nnecke, vosper以及一些其他的學(xué)者在這方面做出了突出的貢獻(xiàn)。本書中包括了這些結(jié)果，并且用freiman定理的ruzsa證明將本書的內(nèi)容推向了高潮。

書籍目錄

prefacenotation and conventiori waring's problem1 sums of polygor1.1 polygonal number1.2 lagrange's theorem1.3 quadratic forms1.4 ternary quadratic forms1.5 sums of three squares1.6 thin sets of squares1.7 the polygonal number theorem1.8 notes1.9 exercises2 waring's problem for cubes2.1 sums of cubes2.2 the wieferich-kempner theorem2.3 linnik's theorem2.4 sums of two cubes2.5 notes.2.6 exercises3 the hilbert-waring theorem3.1 polynomial identities and a conjecture of hurwitz3.2 hermite polynomials and hilbert's identity3.3 a proof by induction3.4 notes3.5 exercises4 weyl's inequality4.1 tools4.2 difference operator4.3 easier waring's problem4.4 fractional parts4.5 weyl's inequality and hua's lemma4.6 notes4.7 exercises5 the hardy-littlewood asymptotic formula5.1 the circle method5.2 waring's problem for k = 15.3 the hardy-littlewood decomposition5.4 the minor arcs5.5 the major arcs5.6 the singular integral5.7 the singular series5.8 conclusion5.9 notes5.10 exercisesii the goldbach conjecture6 elementary estimates for primes6.1 euclid's theorem6.2 chebyshev's theorem6.3 merter's theorems6.4 brun's method and twin primes6.5 notes6.6 exercises7 the shnirel'man-goldbach theorem7.1 the goldbach conjecture7.2 the selberg sieve7.3 applicatior of the sieve7.4 shnirel'man derity7.5 the shnirel'man-goldbach theorem7.6 romanov's theorem7.7 covering congruences7.8 notes7.9 exercises8 sums of three primes8.1 vinogradov's theorem8.2 the singular series8.3 decomposition into major and minor arcs8.4 the integral over the major arcs8.5 an exponential sum over primes8.6 proof of the asymptotic formula8.7 notes8.8 exercise9 the linear sieve9.1 a general sieve9.2 cortruction of a combinatorial sieve9.3 approximatior9.4 the jurkat-richert theorem9.5 differential-difference equatior9.6 notes9.7 exercises10 chen's theorem10.1 primes and almost primes10.2 weights10.3 prolegomena to sieving10.4 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 inequality10.8 conclusion10.9 notesiii appendixarithmetic functiora.1 the ring of arithmetic functiora.2 sums and integralsa.3 multiplicative functiora.4 the divisor functiona.5 the euler φ-functiona.6 the mobius functiona.7 ramanujan sumsa.8 infinite productsa.9 notesa.10 exercisesbibliographyindex

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