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S. Ramanujan

"I was struck by the extraordinary mathematical results contained in it [the notebooks]. I had no mind to smother his genius by an appointment in the lowest rungs of the revenue department" as V.Ramaswami Iyar recalled the situation when S. Ramanujan an Indian Mathematician, showed his math books and wishing for a job at the revenue department where Iyar worked. Ramanujan, with the help of Ramaswami Iyer, had his work published in the `Journal of Indian Mathematical Society`. Ramanujan`s writing initially had many flaws. As Journal editor M. T. Narayana Iyengar noted: "Mr. Ramanujan`s methods were so terse and novel and his presentation so lack in clearness and precision, that the ordinary [mathematical reader], unaccustomed to such intellectual gymnastics, could hardly follow him". With all these he is widely regarded as one of the greatest mathematical minds in recent history. With almost no formal training in pure mathematics, he made substantial contributions in the areas of mathematical analysis, number theory, infinite series and continued fractions. He is a person with a somewhat shy and quiet disposition, dignified with pleasant manners.

S. RamanujanSrinivasa Ramanujan Lyengar, was born on 22nd December 1887 in Erode, Tamilnadu India. When Ramanujan was a year old his mother took him to the town of Kumbakonam, about 160 km nearer Madras. From her, he learned about tradition, the caste system and puranas. His father worked in Kumbakonam as a clerk in a cloth merchant`s shop. In December 1889 he contracted smallpox. He married on 14 July 1909 when his mother arranged for him to marry a ten year old girl S Janaki Ammal. Ramanujan did not live with his wife, however, until she was twelve years old.

At the age of five, Ramanujan entered the primary school in Kumbakonam although he would attend several different primary schools before entering the Town High School in Kumbakonam in January 1898. At the Town High School, Ramanujan was doing well in all his school subjects and showed himself an able all round scholars. He received a scholarship to study at Government College in Kumbakonam known as the "Cambridge of South India." He demonstrated a natural ability at math, and he was given books on advanced trigonometry by S. L. Loney. However, he was so intent on studying mathematics that he could not focus on any other subjects and failed most of them, losing his scholarship in the process. In December 1906 Ramanujan failed his F. A. degree exam. Without a degree, he left college and continued to pursue independent research in mathematics. At this point in his life, he lived in extreme poverty and was often near the point of starvation. He began to work on his own on mathematics summing geometric and arithmetic series. Ramanujan was conducting his own mathematical research on `Bernoulli numbers` and the `Euler-Mascheroni constant`.

Ramanujan sent samples of his theorems to three academics at University of Cambridge, in 1912-1913. Only G. H. Hardy recognized his brilliant work, and he asked Ramanujan to study under him at Cambridge. While working with G.H. Hardy and independently, Ramanujan compiled nearly 3900 results mostly identities and equations during his short lifetime. An identity ia an equality that remains true regardless of the values of any variables that appeare within it. Equation is a mathemetical statement in symbols that two things are same. Most of his claims have now been proven to be correct. He stated results that were both original and highly unconventional, such as the `Ramanujan prime` and the `Ramanujan theta function`, and these have inspired a vast amount of further research. However, some of his major discoveries have been rather slow to enter the mathematical mainstream. Recently, Ramanujan`s formulae have found applications in the field of `crystallography` and in `string theory`.

Ramanujan spent nearly five years in Cambridge collaborating with Hardy and Littlewood and published a part of his findings there. Ramanujan was awarded a B.A. degree in March 1916 for his work on highly composite numbers which was published as a paper in the "Journal of the London Mathematical Society". He was elected to the London Mathematical Society on 6 December 1917. He was elected a Fellow of the Royal Society in 1918 for his investigation in Elliptic Functions and the Theory of Numbers. Thus he became the second Indian to become a Fellow and ofcourse one of the youngest Fellows in the entire history of the Royal Society. On 13 October 1918, he became the first Indian to be elected a Fellow of Trinity College, Cambridge for his contribution in developing the `Number Theory`.

Ramanujan`s health worsened in England as he was diagnosed with tuberculosis and a severe vitamin deficiency and was confined to a sanatorium. He returned to India in 1919 and died soon after at the age of 32. In an analysis of Ramanujan`s medical reports later in 1994 it was revealed that he had a parasitic infection of the liver in medical term hepatic amoebiasis.

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.

Since paper was very expensive, Ramanujan did most of his work on slate (a piece of flat material used as a medium for writing). And then transfer just the results to paper. Ramanujan recorded the bulk of his results in four notebooks of loose leaf paper. These results were mostly written up without any derivations. Thus it was misperceived that Ramanujan was unable to prove his results and simply thought up the final result directly. Mathematician Bruce C. Berndt, in his review of these notebooks and Ramanujan`s work, says that Ramanujan most certainly was able to make the proofs of most of his results, but chose not to.

Ramanujan was considered as the master of theory of numbers. His most outstanding contribution was his formula for p (n), the number of `partitions` of `n`. Ramanujan is generally hailed as an all time great mathematician like Euler, Gauss or Jacobi for his natural genius. G. H. Hardy went on to claim that his greatest contribution to mathematics came from Ramanujan, a fellow and friend quotes on him: "The limitations of his knowledge were as startling as its profundity. Here was a man who could work out modular equations and theorems... to orders unheard of, whose mastery of continued fractions was... beyond that of any mathematician in the world, who had found for himself the functional equation of the zeta function and the dominant terms of many of the most famous problems in the analytic theory of numbers; and yet he had never heard of a doubly-periodic function or of Cauchy`s theorem, and had indeed but the vaguest idea of what a function of a complex variable was..."

Noted physicist Jayant Narlikar in his book "Scientific Edge", stated that "Srinivasa Ramanujan, discovered by the Cambridge mathematician G.H. Hardy, whose great mathematical findings were beginning to be appreciated from 1915 to 1919. His achievements were to be fully understood much later, well after his untimely death in 1920. For example, his work on the highly composite numbers (numbers with a large number of factors) started a whole new line of investigations in the theory of such numbers."

Ramanujan`s work was one of the top ten achievements of 20th century Indian science and could be considered in the Nobel Prize class. Government of India issued a commemorative stamp to honor 75th anniversary birth of Ramanujan. Ramanujan`s home state Tamil Nadu celebrates December 22 Ramanujan`s birthday as `State IT Day`. Indian National Science academy and many other scientific institutions in India are giving various awards in memory of this brilliant mathematician.

An international feature film on Ramanujan`s life produced by an Indo-British collaboration will begin shooting in 2007 in Tamil Nadu state and Cambridge. It will be co-directed by Stephen Fry and Dev Benegal. Another film based on the book `The Man Who Knew Infinity: A Life of the Genius Ramanujan` by Robert Kanigel is being made by Edward Pressman and Matthew Brown.

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