8 月 22 日上午 9:00, 张益唐在中科院数学与系统科学研究院(Academy of Mathematics and Systems Science (AMSS) in the Chinese Academy of Sciences (CAS)) 做了题为 “Prime gaps and related problems” 的讲座, 这是今年中科院的华罗庚数学讲座(Loo-Keng Hua Distinguished Lecture).

Yitang zhang’s Loo-Keng Hua distinguished lecture at CAS 1

Yitang zhang’s Loo-Keng Hua distinguished lecture at CAS 2

Yitang zhang received Loo-Keng Hua souvenir from Yuan Wang

## 质数 $$k$$-tuples 猜想

$\mathcal H=(h_1,h_2,\dotsc,h_{k_0}),$

Hardy-Littlewood prime tuples conjecture  如果 $$k_0$$-tuples $$\mathcal H$$ 是允许的, 那么, 存在无穷多个正整数 $$n$$, 使得 $$n+\mathcal H$$ 全部由质数组成.

## Second Hardy–Littlewood conjecture

1923 年, Hardy 和 Littlewood 发表了一篇论文[1]. 这篇长达 $$70$$ 页, 已经是数论史上的经典, 的论文提出, 对任意整数 $$m,n\geqslant2$$,

$\pi(m+n)\leqslant\pi(m)+\pi(n).$

## 质数 tuples 猜想与第二 Hardy–Littlewood 猜想不能同时成立

1974 年, Ian Richards 和他的博士研究生 Douglas Hensley 指出[2], Hardy-Littlewood 的两个猜想, 是不相容的. 也就是说, 这两个猜想, 至少有一个是不成立的.

$\pi(n+3159)-\pi(n)=447\gt\pi(3159)=446.$

### Annotations

1. 第一部分, 对于质数 tuples 猜想的介绍, 参考了 Tao 的一篇 blog.

### References

1. G. H. Hardy and J. E. Littlewood, On some problems of “partitio numerorum III: On the expression of a number as a sum of primes. Acta Math, 1923, 44: 1–70.
2. D. Hensley and I. Richards, Primes in intervals. Acta Arith. 25 (1974), pp. 375-391.
3. conjectures concerning primes; a discussion of the use of computers in attacking a theoretical problem. Bulletin of the American Mathematical Society 80:3 (1974), pp. 419-438.

1.21 唯一因式分解定理的证明

2.5  五边形与五角数

$p_m(k)=\dfrac{mk(k-1)}2+k.$

2.8  一个不平凡的结论

2.9 什么数恰好有 $$60$$ 个因数?

$$kn = x^2+y^2+1$$

$$n$$ is a odd number, then there exists positive integer $$k\gt0$$ such that $$kn = x^2+y^2+1$$ for some integers $$x,y$$.

with use of the Chinese remainder theorem we have to solve this problem only for power of primes:

suppose that $$n=p_1^{a_1}p_2^{a_2}\dotsm p_k^{a_k}$$, then we know that for each $$i$$, there exist $$x_i, y_i$$ such that $$p_i^{a_i}$$ divides  $$x_i^2+y_i^2+1$$. Now consider these equations:

$X\equiv x_i\pmod {p_i^{a_i}}, i= 1,2,\dotsc,k.$

these equations have solution because of  Chinese remainder theorem.

similarly these equation have solution:

$Y\equiv y_i\pmod {p_i^{a_i}}, i= 1,2,\dotsc,k.$

now $$n$$ divides  $$X^2+Y^2+1.$$

then we can apply hansel’s lemma. Actually we want to show that if for some $$\alpha$$, there exist $$x,y$$ such that $$p^\alpha$$ divides  $$x^2+y^2+1$$, then for  $$\alpha +1$$ such $$x$$  and $$y$$ exist. For this because in case $$\alpha$$, $$p$$ cannot divide both $$x$$  and $$y$$, then we can use hansel for improve $$\alpha$$ to $$\alpha+1.$$

#### References

1. 华罗庚, 数论导引.
2. Hardy, An introducton to the theory of numbers. 有中文本
3. Tom M. Apostol, Introduction to analytic number therory. 有中文本

On 14 May 2013, Mathematician Yitang Zhang claimed that he has proved there are infinitely many prime gaps shorter than 70 million, which was a weak version of the twin prime conjecture.

[Update, May 21, 2013: 张的论文, 全文 $$56$$ 页已经可以在 Annals of Mathematics 的网站看到: Bounded gaps between primes(subscription required). 这文章的主要结果是证明了

$\varliminf_{n\rightarrow\infty}(p_{n+1}-p_n)\lt7\times10^7,$

1. 成就太过突出

2. 用经典方法逆袭, 用弹弓打死了狗熊.

3.张益唐一直坎坷, 一举成名天下知.

## 石破天惊

4 月 17 日, 数学界最富盛名的数学杂志 Annals of Mathematics 的收件箱出现一篇论文. 这论文居然宣称在一个最古老的数学难题孪生质数猜想上取得重大突破. 专家们对作者张益唐感到陌生. 最要命的是, 张其实只是一所普通大学的讲师, 已经 50 好几.

## 筛法

1980 年代后期, IAS 的 Fields Medal 得主 Enrico Bombieri, Toronto大学的 John Friedlander, 和 Rutgers大学的 Henryk Iwaniec 设法修改level of distribution 的定义, 使得这个修订后的参数达到 $$\frac47$$. GPY 的文章在 2005年出笼以后, 研究人员一窝蜂想把这个修改后的 level of distribution 与 GPY的筛法组合起来, 但没有什么成效.

## 张益唐的工作

Goldston认为, 张的筛法, 没有那么强大, 效果也差一点, 但在 GPY 会有一点奇效. 这样一来, 张把 level of distribution 提高到了 $$\frac12+\frac1{584}$$, 这足以使用 Bombieri, Friedlander, 和 Iwaniec 的方法. “新筛法得出了张的惊天动地的结果, 但不太可能证明孪生质数猜想. 即便假定 level of distribution 最好的结果成立, 从 GPY 的方法只能得出有无穷多对质数, 其差不超过 $$16$$.” Goldston 说.

[Update, June 8, 2013: 去年7月3日, 张益唐前往在科罗拉多州立大学音乐系任教的好友, 音乐指挥家齐雅格家中作客. 当时他与齐雅格正准备离家去看排练, 临走前20分钟, 张益唐想到齐家院子后看不请自来的梅花鹿, 顺便抽根烟.

Yitang Zhang

## 张益唐其人

#### References

1. 季理真, 素数不再孤单: 孪生素数和一个执着的数学家张益唐, May 20, 2013.
2. 汤涛, 张益唐和北大数学 78 级, May 19, 2013.
3. Erica Klarreich, Unheralded Mathematician Bridges the Prime Gap, simons foundation, May 19, 2013.
4. Kenneth Chang, Solving a Riddle of Primes, The New Yorks Times, May 20, 2013.
5. Carolyn Y. Johnson, Globe Staff, Obscure University of New Hampshire math professor takes major step toward elusive proof, May 23,2013.
6. Dan Goldston, Zhang’s Theorem on Bounded Gaps Between Primes.
7. Henryk  Iwaniec, a email to Shing-Tung Yau: Subject: Re: Yitang zhang, May 24,2013.
8. Liam O’brien, That figures: Professor who had to work at Subway dazzles world of maths after solving centuries-old prime number riddle, May 21, 2013
9. 唐嘉丽, 张益唐破解千古数学难题, June 6, 2013.
10. Paul Feely, UNH professor solves ancient mathematics riddle, June 2, 2013.