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
Sep 112013
 

Srinivasa Ramanujan(December 22, 1887 – April 26, 1920)的贤名, 不仅仅是因为他出色的数学才能, 罕有的数学直觉, 还因为他在贫瘠的土壤挣扎, 而留下的关于社会制度与教育制度的话题.

关于 Ramanujan(拉马努金) 的生平, 最好的传记应该是 The Man Who Knew Infinity, 中文书名是知无涯者.

今年, Springer 出版了两本关于 Ramanujan 的书. 第一本是 The Mathematical Legacy of Srinivasa Ramanujan, 作者 Murty, M. Ram, Murty, V. Kumar. 另一本是 Krishnaswami Alladi 的 Ramanujan’s Place in the World of Mathematics. 很有意思的是, Springer 在 2010 年出版了一本书 The Legacy of Alladi Ramakrishnan in the Mathematical Sciences

此外, Springer 以 Ramanujan’s Lost Notebook 为书名, 重新出版了拉马努金的笔记. 目前已经推出四卷, 应该还有一卷. 这套书之前的名称是 Ramanujan’s Notebooks.

1997 年, Ramanujan Journal  创刊,用以发表有关’拉马努金钟爱的数学领域’的研究论文.

2011 年 12 月, 拉马努金诞生125 周年来临之际, 印度政府决定每年的 12 月 22 日, 这是拉马努金的诞辰日, 为’国家数学日’. 这是印度总理 Manmohan Singh 在 2011 年 12 月 26 日宣布的.

国际上主要有两项拉马努金为名的奖: SASTRA Ramanujan Prize 和 The Ramanujan Prize. SASTRA Ramanujan Prize 由 Shanmugha Arts, Science, Technology&Research Academy 在2005 年创立, 颁发给在拉马努金做出过贡献的领域做出成绩的不超过 32 岁的杰出数学家.  至于 The Ramanujan Prize是ICTP 于2005年创立的, 颁发给发展中国家的优秀数学家, 获奖者的年龄限制在相对宽松45岁.

Aug 052013
 

For which positive integers \(a, b, c, d\), any natural number \(n\) can be represented as

\[n=ax^2+by^2+cz^2+dw^2,\]

where \(x, y,z,w\) are integers?

Lagrange’s four-square theorem states that \((a,b,c,d)=(1,1,1,1)\) works. Ramanujan proved that there are exactly \(54\) possible choices for \(a, b, c, d\).

For which positive integers \(a, b, c, d\),

\[n=ax^2+by^2+cz^2+dw^2,\]

is solvable in integers \(x, y,z,w\) for all positive integers \(n\) except one number? For example, \(n=x^2+y^2+2z^2+29w^2\) is solvable for all natural number \(n\) except \(14\), \(n=x^2+2y^2+7z^2+11w^2\) and \(n=x^2+2y^2+7z^2+13w^2\) except \(5\).

P.R.Halmos proved that there are exactly \(88\) possible choices for \(a, b, c, d\).

What integers are not in the range of \(a^2+b^2+c^2-x^2\)? Ramanujan also thought about that.

Jun 092012
 

Bertrand’s postulate states that if \(x\geq4\), then there always exists at least one prime \(p\) with\(x<p<2x-2\). A weaker but more elegant formulation is: for every \(x>1\) there is always at least one prime p such that \(x<p<2x\).

In 1919, Ramanujan used properties of  the  Gamma function to give a simple  proof  that  appeared as  a  paper “A proof of Bertrand’s postulate” in the  Journal of the Indian Mathematical Society \( 11: 181–182\).

Let’s begin with the  Chebyshev function :

The  first  Chebyshev  function \(\vartheta(x)\)  is defined with

\begin{equation}\vartheta(x)=\sum_{p\leqslant x}\log p,\end{equation}

The second Chebyshev function \(\psi(x)\) is given by

\begin{equation}\psi(x)=\sum_{i\geqslant 1}\vartheta(x^{\frac1i}),\end{equation}

so

\begin{equation}\log \left[x\right]!=\sum_{i\geqslant 1}\psi(\frac1ix) ,\end{equation}

where \(\left[x\right]\)denotes as usual the greatest integer \(\leqslant x\).

From \((2)\) we have

\begin{equation}\psi(x)-2\psi(\sqrt{x})=\vartheta(x)-\vartheta(x^\frac12)+\vartheta(x^\frac13)-\dotsb,\end{equation}

and from \((3)\)

\begin{equation}\log\left[x\right]!-2\log\left[\frac12x\right]!=\psi(x)-\psi(\frac12x)+\psi(\frac13x)-\dotsb .\end{equation}

Remembering  that \(\vartheta(x)\) and \(\psi(x)\) are steadily increasing functions, from \((4)\) and \((5)\) we get that

\begin{equation}\psi(x)-2\psi(\sqrt{x})\leqslant\vartheta(x)\leqslant\psi(x);\end{equation}

and

\begin{equation}\psi(x)-\psi(\frac12x)\leqslant \log \left[x\right]!-2\log \left[\frac12x\right]!\leqslant\psi(x)-\psi(\frac12x)+\psi(\frac13x). \end{equation}

But it is easy to see that

\begin{equation}\begin{split}\log \Gamma(x)-2\log \Gamma(\frac12x+\frac12)&\leqslant \log \left[x\right]!-2\log \left[\frac12x\right]!\\&\leqslant\log \Gamma(x+1)-2\log \Gamma(\frac12x+\frac12).\end{split}\end{equation}

Now using Stirling’s approximation we deduce from \((8)\) that

\begin{equation}\log \left[x\right]!-2\log \left[\frac12x\right]!<\frac34x, \text{if}  x>0;\end{equation}

and

\begin{equation}\log \left[x\right]!-2\log \left[\frac12x\right]!>\frac23x, \text{if}  x>300.\end{equation}

It follows from \((7)\), \((9)\) and  \((10)\) that

\begin{equation}\psi(x)-\psi(\frac12x)<\frac34x, \text{if}  x>0;\end{equation}

and

\begin{equation}\psi(x)-\psi(\frac12x)+\psi(\frac13x)>\frac23x, \text{if } x>300. \end{equation}

Now changing \(x\) to \(\frac12x\), \(\frac14x\), \(\frac18x\), \(\dotsc\) in \((11)\) and adding up all the results, we obtain

\begin{equation}\psi(x)<\frac32x,\text{if}  x>0.\end{equation}

Again we have

\begin{equation}\begin{split}\psi(x)-\psi(\frac12x)+\psi(\frac13x)&\leqslant\vartheta(x)+2\psi(\sqrt{x})-\vartheta(\frac12x)+\psi(\frac13x)\\&<\vartheta(x)-\vartheta(\frac12x)+\frac12x+3\sqrt{x},\end {split}\end{equation}

in virtue of  \((6)\) and \((13)\).

It  follow from \((12)\) and \((14)\) that

\begin{equation}\vartheta(x)-\vartheta(\frac{1}{2}x)>\frac{1}{6}x-3\sqrt{x}\text{, if }x>300.\end{equation}

But it is obvious that

\begin{equation}\frac16x-3\sqrt{x}\geqslant 0\text{, if } x\geqslant 324,\end{equation}

Hence

\begin{equation}\vartheta(2x)-\vartheta(x)>0, \text{if } x\geqslant162.\end{equation}

In other words there is at least one prime between \(x\) and \(2x\) if  \(x\geq162\). Thus Bertrand’s postulate is proved for all values of  \(x\) not less than \(162\); and, by actual verification, we find that it is true for smaller values.

 

It is obvious that

\begin{equation}\vartheta(x)-\vartheta(\frac12x)\leqslant(\pi(x)-\pi(\frac12x))\log x,\end{equation}

for all values of  \(x\). It follows from \((15)\) and \((18)\) that

\begin{equation}\pi(x)-\pi(\frac12x)>\frac1{\log x}(\frac16x-3\sqrt{x})\text{, if}  x>300.\end{equation}

From this we easily deduce that

\begin{equation}\pi(x)-\pi(\frac12x)\geqslant1,2,3,4,5,\dotsc\text{, if } x\geqslant2,11,17,29,41,\dotsc\end{equation}

respectively.