$$\pi(bq) > \pi(aq)$$

$$\frac{\pi(bq)}{\pi(aq)}\sim\frac{b\ln(aq)}{a\ln(bq)} =\frac{b(\ln q+\ln a)}{a(\ln q+\ln b)}\sim\frac ba>1.$$

$$\pi(p_n)=n\sim\frac{p_n}{\ln p_n},$$

$$\ln p_n\sim\ln\frac{p_n}{\ln p_n}\sim\ln n.$$

$$\lim_{k\to\infty}\frac{p_{n_k}}{p_{m_k}} = \lim_{k\to\infty}\frac{n_k\ln n_k}{m_k\ln m_k}=a$$

7 月他在母校北京大学的北京国际数学研究中心 (BICMR) 有一个系列的学术报告: Distribution of Prime Numbers and the Riemann Zeta Function I, II, III. 这个报告分三场, 原定时间是 July 8, 10, 15,  2014 16:00-17:00, 地点是镜春园 78 号院的 77201 室.

BICMR 官网上这个报告的 Abstract 是这么写的:

The distribution of prime numbers is one of the most important subjects in number theory.

There are many interesting problems in this field. It may not be difficult to understand the problems themselves, but the solutions are extremely difficult.

In this series of talks we will describe the application of certain analytic tools to the distribution of prime numbers. In particular, the role played by the Riemann zeta function will be discussed. We will also describe some early and current researches on the Riemann Hypothesis.

These talks are open to everyone in the major of mathematics, including undergraduate students.

Yitang Zhang at BICMR :Distribution of Prime Numbers and the Riemann Zeta Function

8 日下午 4 点, 田刚现身. 因为人比较多, 改为在镜春园82号甲乙丙楼的中心报告厅进行. 主持人刘若川是 1999 年的 IMO 金牌(他本来也是 1998 年中国国家队的队员).

$\zeta(2k)=\sum_{n=1}^\infty\frac1{n^{2k}}=(-1)^{k+1}\frac{(2\pi)^{2k}B_{2k}}{2(2k)!}$

10 日下午 4 点的第二场, 依旧在镜春园82号甲乙丙楼的中心报告厅. 不过, 15 日的一场会在镜春园 78 号院的 77201 室, 16:30 开始.

15 日下午 4:30 的最后一场, 要深入一点. 田刚坐在教室最后一排, 刘若川, 许晨阳坐在教室左边的走廊.张大师谈到有 Goldston, Pintz and Yildirim 的工作, 说他自己最大的贡献是把 $$c$$ 改进为 $$\dfrac14+\dfrac1{1168}$$.

8月 23 日, 26日上午, 张益唐在晨兴数学中心(Morningside Center of Mathematics, Chinese Academy of Sciences)110 房间以 “Distribution of Prime Numbers” 为题, 做了更细节的讲解.

Yitang Zhang at Morningside Center of Mathematics

23 日

Part 2

Part 3

Part 4

26 日

Part 2

Part 3

Part 4

Yitang Zhang’s Peking University Lecture

Yitang Zhang shaked Chen-Ning Franklin Yang’s hand

8 月 23 日下午 15:00, 张益唐在清华大学主楼三层接待厅做了题为 “Bounded gaps between primes and relevant problems” 的报告, 这是今年清华大学的华罗庚数学讲座(Loo-Keng Hua Distinguished Lecture). 很荣幸, 14:50 过一点点, 我正要踏完主楼的最后几级台阶, 一辆轿车停下来, 一个老人下来了. 这背影好眼熟, 原来是杨振宁! 哎呀, 张益唐的成就过于突出, 把物理学家都吸引来了! 在此, 祝愿杨老健康长寿!

Yitang zhang received souvenir from Yat-Sun Poon

Yitang Zhang’s Tsinghua University Loo-Keng Hua Distinguished Lecture

Yitang Zhang was talking to students at the tea time

Yitang Zhang with students

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

Busy day in analytic number theory

On May 13, 2013, Harald Andres Helfgott  uploaded to the arXiv his paper “Major arcs for Goldbach’s theorem” claimed that he has proved the ternary Goldbach conjecture, or odd Goldbach conjecture, asserts that every odd integer  $$n>5$$ is the sum of three primes.

Goldbach’s conjecture 已经有 $$271$$ 年的历史了.

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.