陈景润学术地位无人可及

本文附件是筛法英文维基的的原文。文中作者把陈景润1+2定理看做是哥德巴赫与孪生素数两个猜想的“近乎无误”（near-misses）的数学证明，陈景润的学术地位比数学家比龙还要高一些。

请读者仔细阅读本文附件，便可知一切，

袁萌陈启清 2月6日

附件：以下是筛法英文维基的原文

Metaphor（隐喻）: Various physical sieves（物理筛子）

Sieve theory is a set of general techniques in number theory, designed to count, or more realistically to estimate the size of, sifted sets of integers. The prototypical example of a sifted set is the set of prime numbers up to some prescribed limit X. Correspondingly, the prototypical example of a sieve is the sieve of Eratosthenes, or the more general Legendre sieve. The direct attack on prime numbers using these methods soon reaches apparently insuperable obstacles, in the way of the accumulation of error terms.[citation needed] In one of the major strands of number theory in the twentieth century, ways were found of avoiding some of the difficulties of a frontal attack with a naive idea of what sieving should be.[citation needed]

One successful approach is to approximate a specific sifted set of numbers (e.g. the set of prime numbers) by another, simpler set (e.g. the set of almost prime numbers), which is typically somewhat larger than the original set, and easier to analyze. More sophisticated sieves also do not work directly with sets per se, but instead count them according to carefully chosen weight functions on these sets (options for giving some elements of these sets more "weight" than others). Furthermore, in some modern applications, sieves are used not to estimate the size of a sifted set, but to produce a function that is large on the set and mostly small outside it, while being easier to analyze than the characteristic function of the set.

ontents

1 Types of sieving

2 Techniques of sieve theory

3 References

4 External links

Types of sievingnea

Modern sieves include the Brun sieve, the Selberg sieve, the Turán sieve, the large sieve, and the larger sieve. One of the original purposes of sieve theory was to try to prove conjectures in number theory such as the twin prime conjecture. While the original broad aims of sieve theory still are largely unachieved, there have been some partial successes, especially in combination with other number theoretic tools. Highlights include:

Brun's theorem, which asserts that the sum of the reciprocals of the twin primes converges (whereas the sum of the reciprocals of the primes themselves diverges);

Chen's theorem, which shows that there are infinitely many primes p such that p + 2 is either a prime or a semiprime (the product of two primes); a closely related theorem of Chen Jingrun asserts that every sufficiently large even number is the sum of a prime and another number which is either a prime or a semiprime. These can be considered to be near-misses to the twin prime conjecture and the Goldbach conjecture respectively.

The fundamental lemma of sieve theory, which (very roughly speaking) asserts that if one is sifting a set of N numbers, then one can accurately estimate the number of elements left in the sieve after

N ε {\displaystyle N^{\varepsilon }}

iterations provided that

ε {\displaystyle \varepsilon }

is sufficiently small (fractions such as 1/10 are quite typical here). This lemma is usually too weak to sieve out primes (which generally require something like

N 1 / 2 {\displaystyle N^{1/2}}

iterations), but can be enough to obtain results regarding almost primes.

The Friedlander–Iwaniec theorem, which asserts that there are infinitely many primes of the form

a 2 + b 4 {\displaystyle a^{2}+b^{4}}

Zhang's theorem (Zhang 2014) that there are infinitely many pairs of primes within a bounded distance. The Maynard–Tao theorem (Maynard 2015) generalizes Zhang's theorem to arbitrarily long sequences of primes.

Techniques of sieve theory[edit]

The techniques of sieve theory can be quite powerful, but they seem to be limited by an obstacle known as the parity problem, which roughly speaking asserts that sieve theory methods have extreme difficulty distinguishing between numbers with an odd number of prime factors and numbers with an even number of prime factors. This parity problem is still not very well understood.

Compared with other methods in number theory, sieve theory is comparatively elementary, in the sense that it does not necessarily require sophisticated concepts from either algebraic number theory or analytic number theory. Nevertheless, the more advanced sieves can still get very intricate and delicate (especially when combined with other deep techniques in number theory), and entire textbooks have been devoted to this single subfield of number theory; a classic reference is (Halberstam & Richert 1974) and a more modern text is (Iwaniec & Friedlander 2010).

The sieve methods discussed in this article are not closely related to the integer factorization sieve methods such as the quadratic sieve and the general number field sieve. Those factorization methods use the idea of the sieve of Eratosthenes to determine efficiently which members of a list of numbers can be completely factored into small primes.

References[edit]

This article includes a list of references, related reading or external links, but its sources remain unclear because it lacks inline citations. Please help to improve this article by introducing more precise citations. (July 2009) (Learn how and when to remove this template message)

Cojocaru, Alina Carmen; Murty, M. Ram (2006), An introduction to sieve methods and their applications, London Mathematical Society Student Texts, 66, Cambridge University Press, ISBN 0-521-84816-4, MR 2200366

Motohashi, Yoichi (1983), Lectures on Sieve Methods and Prime Number Theory, Tata Institute of Fundamental Research Lectures on Mathematics and Physics, 72, Berlin: Springer-Verlag, ISBN 3-540-12281-8, MR 0735437

Greaves, George (2001), Sieves in number theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 43, Berlin: Springer-Verlag, doi:10.1007/978-3-662-04658-6, ISBN 3-540-41647-1, MR 1836967

Harman, Glyn (2007). Prime-detecting sieves. London Mathematical Society Monographs. 33. Princeton, NJ: Princeton University Press. ISBN 978-0-691-12437-7. MR 2331072. Zbl 1220.11118.

Halberstam, Heini; Richert, Hans-Egon (1974). Sieve Methods. London Mathematical Society Monographs. 4. London-New York: Academic Press. ISBN 0-12-318250-6. MR 0424730.

Iwaniec, Henryk; Friedlander, John (2010), Opera de cribro, American Mathematical Society Colloquium Publications, 57, Providence, RI: American Mathematical Society, ISBN 978-0-8218-4970-5, MR 2647984

Hooley, Christopher (1976), Applications of sieve methods to the theory of numbers, Cambridge Tracts in Mathematics, 70, Cambridge-New York-Melbourne: Cambridge University Press, ISBN 0-521-20915-3, MR 0404173

Maynard, James (2015). "Small gaps between primes". Annals of Mathematics. 181 (1): 383–413. arXiv:1311.4600. doi:10.4007/annals.2015.181.1.7. MR 3272929.

Tenenbaum, Gérald (1995), Introduction to Analytic and Probabilistic Number Theory, Cambridge studies in advanced mathematics, 46, Translated from the second French edition (1995) by C. B. Thomas, Cambridge University Press, pp. 56–79, ISBN 0-521-41261-7, MR 1342300

Zhang, Yitang (2014). "Bounded gaps between primes". Annals of Mathematics. 179 (3): 1121–1174. doi:10.4007/annals.2014.179.3.7. MR 3171761.

External links[edit]

Bredikhin, B.M. (2001) [1994], "Sieve method", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4

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