Aug 212014
 

Yitang Zhang is giving the last invited talk at ICM 2014, “Small gaps between primes and primes in arithmetic progressions to large moduli”.

Yitang Zhang is giving the last invited talk

Yitang Zhang is giving the last invited talk 1

这是闭幕式前的最后一个 invited talk. 张大师习惯手写, 当场演算.

Yitang Zhang stepped onto the main stage of mathematics last year with the announced of his achievement that is hailed as “a landmark  theorem in the distribution of prime numbers”.

Yitang Zhang is giving the last invited talk

Yitang Zhang is giving the last invited talk 2

Yitang Zhang is giving the last invited talk

Yitang Zhang is giving the last invited talk 3

Aug 052014
 

ICM 2014 Program
这届国际数学界大会(International Congress of Mathematicians, ICM)的安排, 已经明确无误的说明:

 4 个数学家将获得本届大会的 Fields Medal.

张大师将于 8 月 21 日作 ICM 闭幕式之前的压轴报告, 这是只有今年的 Fields Medalist, Gauss Prize, Chern Medal 得主才有的殊荣

张益唐 7 月1 日在北大本科生毕业典礼有一个讲话

这个暑假, 张大师在中国科学院晨兴数学中心和他的母校北京大学做了好几次讲座.

1. A Transition Formula for Mean Values of Dirichlet Polynomials
2014,6.23./6.25. 9:30-11:30
晨兴 110
主持人: 王元

2. 关于 Siegel 零点
2014.7.2.9:30-11:30
晨兴 110

3. Distribution of Prime Numbers and the Riemann Zeta Function
July 8, 10, 2014 16:00-17:00, 镜春园82号甲乙丙楼的中心报告厅
July 15, 16:30-17:30 镜春园 78 号院的 77201 室.
主持人: 刘若川

4. 关于 Siegel 零点(2)
014.7.16./7.30./8.4./8.6. 9:30-11:30
N820

Jul 292014
 

I’ve just received a book named Number Theory in the Spirit of Liouville by Kenneth S. Williams.

Number Theory in the Spirit of Liouville

Number Theory in the Spirit of Liouville

Joseph Liouville is recognised as one of the great mathematicians of the nineteenth century, and one of his greatest achievements was the introduction of a powerful new method into elementary number theory. This book provides a gentle introduction to this method, explaining it in a clear and straightforward manner. The many applications provided include applications to sums of squares, sums of triangular numbers, recurrence relations for divisor functions, convolution sums involving the divisor functions, and many others. All of the topics discussed have a rich history dating back to Euler, Jacobi, Dirichlet, Ramanujan and others, and they continue to be the subject of current mathematical research. Williams places the results in their historical and contemporary contexts, making the connection between Liouville’s ideas and modern theory. This is the only book in English entirely devoted to the subject and is thus an extremely valuable resource for both students and researchers alike.

  • Demonstrates that some analytic formulae in number theory can be proved in an elementary arithmetic manner
  • Motivates students to do their own research
  • Includes an extensive bibliography

Table of Contents

Preface
1. Joseph Liouville (1809–1888)
2. Liouville’s ideas in number theory
3. The arithmetic functions \(\sigma_k(n)\), \(\sigma_k^*(n)\), \(d_{k,m}(n)\) and \(F_k(n)\)
4. The equation \(i^2+jk = n\)
5. An identity of Liouville
6. A recurrence relation for \(\sigma^*(n)\)
7. The Girard–Fermat theorem
8. A second identity of Liouville
9. Sums of two, four and six squares
10. A third identity of Liouville
11. Jacobi’s four squares formula
12. Besge’s formula
13. An identity of Huard, Ou, Spearman and Williams
14. Four elementary arithmetic formulae
15. Some twisted convolution sums
16. Sums of two, four, six and eight triangular numbers
17. Sums of integers of the form \(x^2+xy+y^2\)
18. Representations by \(x^2+y^2+z^2+2t^2\), \(x^2+y^2+2z^2+2t^2\) and \(x^2+2y^2+2z^2+2t^2\)
19. Sums of eight and twelve squares
20. Concluding remarks
References
Index.

Review

“… a fascinating exploration and reexamination of both Liouville’s identities and “elementary” methods, providing revealing connections to modern techniques and proofs. Overall, the work contributes significantly to both number theory and the history of mathematics.”

J. Johnson, Choice Magazine

Publisher: Cambridge University Press (November 29, 2010)
Language: English
FORMAT: Paperback
ISBN: 9780521175623
LENGTH: 306 pages
DIMENSIONS: 227 x 151 x 16 mm
CONTAINS: 275 exercises

Jul 272014
 

I’ve just received a book named Development of Elliptic Functions According to Ramanujan
by K Venkatachaliengar (deceased) , edited by: ShaunCooper , Shaun Cooper

Development of Elliptic Functions According to Ramanujan

Development of Elliptic Functions According to Ramanujan

This unique book provides an innovative and efficient approach to elliptic functions, based on the ideas of the great Indian mathematician Srinivasa Ramanujan. The original 1988 monograph of K Venkatachaliengar has been completely revised. Many details, omitted from the original version, have been included, and the book has been made comprehensive by notes at the end of each chapter.

The book is for graduate students and researchers in Number Theory and Classical Analysis, as well for scholars and aficionados of Ramanujan’s work. It can be read by anyone with some undergraduate knowledge of real and complex analysis.

Contents:

  • The Basic Identity
  • The Differential Equations of \(P\), \(Q\) and \(R\)
  • The Jordan-Kronecker Function
  • The Weierstrassian Invariants
  • The Weierstrassian Invariants, II
  • Development of Elliptic Functions
  • The Modular Function \(\lambda\)

Readership: Graduate students and researchers in Number Theory and Classical Analysis, as well as scholars and aficionados of Ramanujan’s work.

Review

It is obvious that every arithmetician should want to own a copy of this book, and every modular former should put it on his ‘to be handled-with-loving-care-shelf.’ Reader of Venkatachaliengar’s fine, fine book should be willing to enter into that part of the mathematical world where Euler, Jacobi, and Ramanujan live: beautiful formulas everywhere, innumerable computations with infinite series, and striking manouevres with infinite products.

— MAA Reviews

The author was acquainted with many who knew Ramanujan, and so historical passages offer information not found in standard biographical sources. The author has studied Ramanujan’s papers and notebooks over a period of several decades. His keen insights, beautiful new theorems, and elegant proofs presented in this monograph will enrich readers.

— MathSciNet

The author has studied Ramanujan’s papers and notebooks over a period of several decades. His keen insights, beautiful new theorems, and elegant proofs presented in this monograph will enrich readers. italic Zentralblatt MATH

— Zentralblatt MATH

  • Series: Monographs in Number Theory (Book 6)
  • Hardcover: 184 pages
  • Publisher: World Scientific Publishing Company (September 28, 2011)
  • Language: English
  • ISBN-10: 9814366455
  • ISBN-13: 978-9814366458
Mar 252014
 

Which integers can be expressed as \(a^3+b^3+c^3-3abc\)? \(a\), \(b\), \(c\in\Bbb Z\).

\[(a\pm1)^3+a^3+a^3-3(a\pm1)a^2=3a\pm1\]

\[(a-1)^3+a^3+(a+1)^3-3a(a+1)(a-1)=9a\]

\[2(a^3+b^3+c^3-3abc)=3(a+b+c)(a^2+b^2+c^2)-(a+b+c)^3\]

If \(3\mid(a^3+b^3+c^3-3abc)\), then \(3\mid(a+b+c)^3\), \(3\mid(a+b+c)\). so \(9\mid(a^3+b^3+c^3-3abc)\).

All \(n\) such that \(3\nmid n\) or \(9\mid n\).

Feb 112014
 

Acta Arithmetica(ISSN: 0065-1036(print) 1730-6264(online)) is a scientific journal of mathematics publishing papers on number theory. It was established in 1935 by Salomon Lubelski and Arnold Walfisz. The journal is published by the Institute of Mathematics of the Polish Academy of Sciences.

1935 年, Salomon Lubelski 和 Arnold Walfisz 创立了Acta Arithmetica.

Acta Arithmetica 是一个数学杂志, 发表数论方面的原创研究论文, 由 Polish(波兰)科学院的数学研究所出版. 从 1995 年开始, Acta Arithmetica 每年出版 5 卷(2012 年有 6 卷; 1996-2000 年间, 每年 4.5 卷), 刊登 80-100 篇论文.

目前, Acta Arithmetica 第 1-95 卷是 Open Access(开放存取), 而第 96 卷以及第 96 卷之后, 读者需要订阅才可以看到全文. 因为第 96 卷的后 2 期在 2001 年刊发, 因此, 任何人都可以及时, 免费, 不受任何限制地通过网络获取 2000 年以及之前的所有 Acta Arithmetica, 除了 2000 年最后的第 96 卷的前 2 期.

Feb 042014
 

Let \(f(x)=x^n+a_{n-1}x^{n-1} +\dotsb+a_1x+a_0\) be a polynomial with integer coefficients, and let \(d_1\),\(\dotsc\), \(d_n\) be pairwise distinct integers. Suppose that for infinitely many prime numbers \(p\) there exists an integer \(k_p\) for which

\begin{equation}f\left(k_p+d_1\right)\equiv f\left(k_p+d_2\right)\equiv\dotsb\equiv f\left(k_p+d_n\right)\equiv0\pmod p.\end{equation}

Prove that there exists an integer \(k_0\) such that

\[f\left(k_0+d_1\right)=f\left(k_0+d_2\right)=\dotsb= f\left(k_0+d_n\right)=0.\]

用 \(P\) 表示 \(\gt n\), 且具有下列性质的质数 \(p\) 所组成的集合: 存在整数 \(k_p\), 使得 \((1)\) 为真.

记 \(u=d_1+d_2 +\dotsb+d_n+a_{n-1}\). 对于 \(p\in P\), 设 \(K_p=nk_p+u\); 对每个 \(\leq n\) 的正整数 \(i\), 设 \(D_i=nd_i-u\). 易见, 所有的 \(\mid D_i-D_j\mid\) 都不会是 \(0\).

\[F(x)=n^nf\Big(\frac xn\Big)=x^n+na_{n-1}x^{n-1}+n^2a_{n-2}x^{n-2}\dotsb+n^{n-1}a_1x+n^na_0.\]

注意到, 对正整数 \(i\)(\(1\leq i\leq n\)), 有

\[F\big(K_p+D_i\big)=n^nf\bigg(\frac{K_p+D_i}n\bigg)=n^nf\big(k_p+d_i\big)\equiv0\pmod p.\]

只要质数 \(p\in P\) 足够大, 任意的 \(\mid D_i-D_j\mid\) 都不被 \(p\) 整除. 既然 \(K_p+D_1\), \(K_p+D_2\), \(\dotsc\), \(K_p+D_n\) \(\bmod p\) 互不同余, 从而它们就是 \(F(x)\equiv0\pmod p\) 的全部解. 然后, Vieta formula 定出

\[\big(K_p+D_1\big)+\big(K_p+D_2\big)+\dotsb+\big(K_p+D_n\big)\equiv-na_{n-1}\pmod p,\]

\[\big(K_p+nd_1-u\big)+\big(K_p+nd_2-u\big)+\dotsb+\big(K_p+nd_n-u\big)\equiv-na_{n-1}\pmod p,\]

进而

\[nK_p\equiv-n\big(a_{n-1}+d_1+d_2+\dotsb+d_n-u\big)=0\pmod p.\]

既然 \(p\gt n\), \(p\mid K_p\).

再次使用 Vieta formula, 当 \(1\leq l\leq n\) 时,

\[(-1)^ln^la_{n-l}\equiv\prod_{1\leq i_1\lt\dotsb\lt i_l\leq n}\big(K_p+D_{i_1}\big)\dotsm\big(K_p+D_{i_l}\big)\pmod p,\]

由 \(p\mid K_p\) 得知

\begin{equation}(-1)^ln^la_{n-l}\equiv\prod_{1\leq i_1\lt\dotsb\lt i_l\leq n}D_{i_1}\dotsm D_{i_l}\pmod p.\end{equation}

只要 \(P\) 中的质数 \(p\) 使得任意的 \(\mid D_i-D_j\mid\) 都不被 \(p\) 整除, 则 \((2)\) 成立. 如此, 就必须有

\[(-1)^ln^la_{n-l}=\prod_{1\leq i_1\lt\dotsb\lt i_l\leq n}D_{i_1}\dotsm D_{i_l}.\]

从而

\[F(x)=\big(x-D_1\big)\big(x-D_2\big)\dotsm\big(x-D_n\big).\]

返回到多项式 \(f\) 以及 \(d_i\),

\[f(x)=\Big(x-d_1-\frac un\Big)\Big(x-d_2-\frac un\Big)\dotsm\Big(x-d_n-\frac un\Big).\]

\(f\) 是首一多项式, 其有理根必是整数. 故 \(\dfrac un\) 是整数.

令 \(k_0=\dfrac un\), 则 \(f(k_0+d_i)=0\) 对所有的 \(1\leq i\leq n\).