陈景润与华罗庚学派

《堆垒素数论》是1940年（民国二十九年），国立西南联合大学的教授华罗庚在一个吊脚楼上，用八个月完成了第一部数学专著。本书成书于1940-1941年（一说1939-1941年）间，最初投交苏联科学院发表。但由于1941-1945的战争条件，延至1947年在苏联以俄文出版，后来于1953由中国科学院出版中文版。

  1954年期间，陈景润存厦门大学数学系资料室担任资料员 阅读华罗庚《堆垒素数论》，深入研究其中的“塔内问题”，次年发表论文“关于塔内问题”（组合数学），引起华罗庚的关注，于1956年邀请陈景润赴京参加一次数学会议，报告“塔内问题”。

  1957年9月，陈景润调入中科院数额所，跟随华罗庚研究堆垒素数论。1973年，陈景润定理发表，成为“堆垒素数论”的最佳代表作。从此，华罗庚V堆垒素数论学派名扬四海！

袁萌  陈启清   2月11日

附件：陈景润在堆垒素数论中的。地位可见

Serdica Math. J. 31 (2005), 1–74

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.)

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

An invitation to additive prime number theory 3