Friday, February 5, 2010

Contributions of S. Ramanujan

The great Indian mathematician, S. Ramanujan has left the sign of his brilliance in various fields of mathematics like Algebra, Geometry, Trigonometry, Calculus, Number theory etc. throughout his entire life. He has also made some extraordinary contributions to the fields like Hypergeometric series, Elliptic functions, Prime numbers, Bernoulli`s numbers, Divergent series, Continued fractions, Elliptic Modular equations, Highly Composite numbers,
Riemann Zeta functions, Partition of numbers, Mock-Theta functions etc. Actually, apart from a few elementary ones, most of the contributions of S. Ramanujan belong to a higher realm of mathematics that is often referred to as "Higher Mathematics". In fact, one can find it quite difficult to understand S. Ramanujan`s mathematics if he does not have the basic foundation in various mathematical subjects.

According to an eminent mathematician, all the numbers were actually the intimate friends of S. Ramanujan. Ramanujan was so close to the numbers that he made the number 1729 as the `Ramanujan number`, as the other mathematicians call it so in his honour. The main reason behind this is that S. Ramanujan gave its fine characteristics in an anecdote involving G. H. Hardy, who had visited him in a sanatorium by hiring a taxi having this number. In order to calculate the value of pi (.*) up to 17 million places using a computer, the present day mathematicians actually use S. Ramanujan`s fastest step-by-step method. The mathematical contributions of S. Ramanujan have also been widely used in solving various problems in higher scientific fields of specialisation. The diverse specialised higher scientific fields include the likes of particle physics, statistical mechanics, computer science, space science, cryptology, polymer chemistry and medical science. The strange thing is that some of these fields were not even in existence during his lifetime. Apart from the above fields, S. Ramanujan`s mathematical methods are being used in designing better blast furnaces for smelting metals and splicing telephone cables for communications, as well.

S. Ramanujan was highly honoured and respected by all the other mathematicians of the world in his time and some of them openly praised S. Ramanujan as the genius. G. H. Hardy was one of the most prominent among them and it should also be said that Hardy was the man, who had the credit of introducing S. Ramanujan to the whole world. Hardy had an informal scale of rating eminent mathematicians and once in that scale of 100, Hardy gave himself 25, the other mathematicians like Littlewood 30, the German David Hilbert 80, and S. Ramanujan, a full 100. In fact, there was no rival for Ramanujan in the whole world, during his lifetime, as no mathematician was this much dependent upon his native intelligence as he was. Hardy was also quite happy for having the credit of introducing Ramanujan to the world, as he once said his greatest contribution to mathematics was the discovery of Ramanujan. He also later wrote about Ramanujan in his popular book "A Mathematician`s Apology".

S. Ramanujan actually belonged to the `Formalist` school of mathematics, just like the earlier great mathematicians Leonard Euler and Carl Jacobi. Though Ramanujan did not give any serious thought to the deeper meaning of mathematics, he gave a form to mathematics through his formulas, theorems, identities etc. He also searched for forms or patterns in mathematics and he actually worked more by intuition and induction and showed relationships between numbers, something that nobody could even imagine at that time. As he was an untrained mathematician, he never gave importance to stringent proof, which is the hallmark of western mathematics. He just went ahead and built a relationship between numbers without bothering about any proof, whenever, he intuitively and by induction felt something was right. As a result, he also made mistakes and some of his mathematical work was later proved to be either incorrect or lacking sufficient proof. S. Ramanujan had no distinct philosophy about mathematics and he believed that all the finite numbers are nothing but only the products of zero and infinity. S. Ramanujan also stated results that were both original and highly unconventional, like the `Ramanujan prime` and the `Ramanujan theta function`, and these have inspired a vast amount of further research.

S. Ramanujan wrote down all his mathematical findings, in ledger-like notebooks that lately became famous as "Ramanujan`s Frayed Notebooks". All those findings were actually a treasure trove of creative mathematics and the Notebooks contain about 4,000 theorems, formulas, corollaries and examples. According to G. H. Hardy, at least two-third of the mathematics in the notebooks was totally novel, and none of the western mathematicians touched them before. There are actually three notebooks of S. Ramanujan. The `First Notebook` is about 134 pages long and the book is divided into 16 chapters. The `Second Notebook` is actually a revised and enlarged version of the `First Notebook` and the Notebook is about 252 pages long and divided into 21 chapters. The last or the `Third Notebook` is 33 pages long and this book contains all the un organised material of S. Ramanujan`s works. This book was written by Ramanujan during his last days of illness in India and the book is also referred to as the "Lost Notebook". The book was taken as `lost` for some years and it was the American mathematician, George Andrews, of Pennsylvania State University, who eventually discovered the book in 1976. He discovered the book in a box, along with some bills and letters in the library of the Trinity College, Cambridge.

However, none of the `Frayed Notebooks` of S. Ramanujan became popular during his lifetime and they became famous only after his death. The Cambridge University Press brought out his Collected Papers in 1927 and after this, the mathematicians all over the world became fascinated by his work and personality. In 1929, two British mathematicians, G. N. Watson of the University of Birmingham and B. M. Wilson of Liverpool University, started to study and edit the notebooks with an intention to unearth the mathematical gems lying undiscovered in them. However, they could not complete the task, as one of the mathematicians died within a few years. Eventually, the Tata Institute of Fundamental Research, Mumbai published the facsimile editions of the two volumes of Ramanujan`s notebooks in 1957. The books were published at the initiative of the Indian nuclear physicist Homi J. Bhabha and the publication once again, renewed the mathematicians` interest in Ramanujan`s notebooks.

The most celebrated application of the `Ramanujan conjecture` is the explicit construction of Ramanujan graphs by Lubotzky, Phillips and Sarnak. In fact, this conjecture gave a name to the graphs. Although there are numerous statements that could bear the name `Ramanujan conjecture`, there is one statement that was very influential on later work. In particular, the connection of this conjecture with conjectures of A. Weil in algebraic geometry opened up new areas of research. Ramanujan was also considered as the master of numbers. His most outstanding contribution was his formula for p (n), the number of `partitions` of `n`. For all these reasons, Ramanujan is hailed as an all time great mathematician like Euler, Gauss or Jacobi for his natural genius.

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