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

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

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.

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.

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.

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

$$\vartheta(x)=\sum_{p\leqslant x}\log p,$$

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

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

so

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

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

From $$(2)$$ we have

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

and from $$(3)$$

$$\log\left[x\right]!-2\log\left[\frac12x\right]!=\psi(x)-\psi(\frac12x)+\psi(\frac13x)-\dotsb .$$

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

$$\psi(x)-2\psi(\sqrt{x})\leqslant\vartheta(x)\leqslant\psi(x);$$

and

$$\psi(x)-\psi(\frac12x)\leqslant \log \left[x\right]!-2\log \left[\frac12x\right]!\leqslant\psi(x)-\psi(\frac12x)+\psi(\frac13x).$$

But it is easy to see that

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

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

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

and

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

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

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

and

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

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

$$\psi(x)<\frac32x,\text{if} x>0.$$

Again we have

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

in virtue of  $$(6)$$ and $$(13)$$.

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

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

But it is obvious that

$$\frac16x-3\sqrt{x}\geqslant 0\text{, if } x\geqslant 324,$$

Hence

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

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

$$\vartheta(x)-\vartheta(\frac12x)\leqslant(\pi(x)-\pi(\frac12x))\log x,$$

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

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

From this we easily deduce that

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

respectively.