陈景润定理的数学证明何处寻？

由于时代过于久远，陈景润定理的数学证明与公式推理过程很难寻找。

实际上，陈景润定理的数学证明与公式推导十分复、困难，出乎一般人的想象。  

有兴趣者，可搜索该文PDF原文第，查看第5-6页。该文件共有74页，参考文献共有254篇，十分丰富，值得保留；此文发表于2005年。

袁萌  陈启清   2月10日

附件：堆垒素数理论引论原文：

AN INVITATION TO ADDITIVE PRIME NUMBER THEORY

A. V. Kumchev, D. I. Tolev

Communicated by V. Drensky

Abstract. The main purpose of this survey is to introduce the inexperienced reader to additive prime number theory and some related branches of analytic number theory. We state the main problems in the eld, sketch their history and the basic machinery used to study them, and try to give a representative sample of the directions of current research.

1. Introduction. Additive number theory is the branch of number theory that studies the representations of natural numbers as sums of integers subject to various arithmetic restrictions. For example, given a sequence of integers

A = {a1 < a2 < a3 < ···} 2000 Mathematics Subject Classication: 11D75, 11D85, 11L20, 11N05, 11N35, 11N36, 11P05, 11P32, 11P55. Key words: Goldbach problems, additive problems, circle method, sieve methods, prime numbers.

2 A. V. Kumchev, D. I. Tolev

one often asks what natural numbers can be represented as sums of a xed number of elements of A; that is, for any xed s ∈ N, one wants to nd the natural numbers n such that the diophantine equation x1 +···+ xs = n(1.1) has a solution in x1,...,xs ∈ A. The sequence A may be described in some generality (say, one may assume that A contains “many” integers), or it may be a particular sequence of some arithmetic interest (say, A may be the sequence of kth powers, the sequence of prime numbers, the values taken by a polynomial F(X) ∈ Z[X] at the positive integers or at the primes, etc.). In this survey, we discuss almost exclusively problems of the latter kind. The main focus will be on two questions,

known as Goldbach’s problem and the Waring–Goldbach problem, which are concerned with representations as sums of primes and powers of primes, respectively.

1.1.             Goldbach’s problem.

Goldbach’s problem appeared for the rst time in 1742 in the correspondence between Goldbach and Euler. In modern language, it can be stated as follows. Goldbach Conjecture. Every even integer n ≥ 4 is the sum of two primes, and every odd integer n≥ 7 is the sum of three primes. The two parts of this conjecture are known as the binary Goldbach problem and the ternary Goldbach problem, respectively. Clearly, the binary conjecture is the stronger one. It is also much more dicult. The rst theoretical evidence in support of Goldbach’s conjecture was obtained by Brun [27], who showed that every large even integer is the sum of two integers having at most nine prime factors. Brun also obtained an upper bound of the correct order for the number of representations of a large even integer as the sum of two primes. During the early 1920s Hardy and Littlewood [67]–[72] developed the ideas in an earlier paper by Hardy and Ramanujan [73] into a new analytic method in additive number theory. Their method is known as the circle method. In 1923 Hardy and Littlewood [69, 71] applied the circle method to Goldbach’s problem. Assuming the Generalized Riemann Hypothesis1 (GRH), they proved that all but nitely many odd integers are sums of three primes and that all but Ox1/2+ε even integers n ≤ x are sums of two primes. (Henceforth, ε denotes a positive number which can be chosen arbitrarily small if the implied constant is allowed to depend on ε.)

1 An important conjecture about certain Dirichlet series; see §2.2 for details.

An invitation to additive prime number theory 3

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