Srinivasa Aiyangar Ramanujan (December 22, 1887 – April 26, 1920) was an Indian mathematician and one of the most esoteric mathematical geniuses in the twentieth century. Nicknamed as "the man who knew infinity", who had uncanny mathematical manipulative abilities. He excelled in number theory and modular functions. He also made significant contributions to the development of partition functions and summation formulas involving constants such as π. A child prodigy, he was largely self-taught in mathematics and had compiled over 3,000 theorems by the year 1914 when he moved to Cambridge. Often, his formulas were stated without proof and were only later proven to be true. His results have inspired a large amount of research and mathematical papers. In 1997 the Ramanujan Journal was launched to publish work "in areas of mathematics influenced by Ramanujan".

Life

Childhood and early life

Ramanujan was born in 1887 in Erode, Tamil Nadu, India. In 1898 at age 10, he entered the Town High School in Kumbakonam, where he appears to have first encountered formal mathematics. At 11 he had mastered the mathematical knowledge of the lodgers at his home, both students at the Government College, and was lent books on advanced trigonometry, by S.L.Loney, which he mastered by 13. His biographer reports that by 14 his genius was beginning to show. Not only did he achieve merit certificates and academic awards throughout his school years, but was also assisting the school in the logistics of assigning its 1200 students (each with their own needs) to its 35-odd teachers, completing mathematical exams in half the allotted time, and already showing familiarity with infinite series. His peers at the time later commented "We, including teachers, rarely understood him" and "stood in respectful awe" of him. However, Ramanujan could not concentrate on other subjects and failed his high school exams. At this time in his life, he was also quite poor and was often pushed to the point of starvation.

Adulthood in India

After marriage, he began his search for a job. With the packet of his mathematical calculations, he moved around in the city of Chennai on the look out for a clerical job. He finally got one at the Chennai Accountant General's Office. Ramanujan desired the luxury to completely focus on mathematics, and was advised by an Englishman to contact researchers in Cambridge. He doggedly solicited support from influential Indian individuals and published several papers in Indian mathematical journals, but was unsuccessful in his attempts to foster sponsorship. It was at this point in time that Sir Ashutosh Mukherjee tried to support his cause.

In late 1912 and early 1913 Ramanujan sent letters and examples of his theorems to three Cambridge academics: H. F. Baker, E. W. Hobson, and G. H. Hardy. Only Hardy, a Fellow of Trinity College to whom Ramanujan wrote in January 1913, noticed the genius demonstrated by the theorems.

Upon reading the initial unsolicited missive by an unknown and untrained Indian mathematician, Hardy and his colleague J.E. Littlewood commented that, “not one could have been set in the most advanced mathematical examination in the world.” Although Hardy was one of the pre-eminent

Life in England

After some initial skepticism, Hardy replied with comments, requesting proofs for some of the discoveries, and began to make plans to bring Ramanujan to England. As an orthodox Brahmin, Ramanujan consulted the astrological data for his journey, because of religious concerns that he would lose his caste by traveling to foreign shores. Ramanujan's mother had a dream in which the family Goddess told her not to stand in the way of her son's travel, and so he made plans accordingly, although he took pains to keep a proper Brahmin lifestyle as far as he could, when he did.

Hardy said of Ramanujan's formulae, some of which he could not initially understand, that "a single look at them is enough to show that they could only be written down by a mathematician of the highest class. They must be true, for if they were not true, no one would have had the imagination to invent them." Hardy stated in an interview by Paul Erdős that his own greatest contribution to mathematics was the discovery of Ramanujan, and compared Ramanujan at least to the mathematical giants Euler and Jacobi in terms of genius. Ramanujan was later appointed a Fellow of Trinity, and a Fellow of the Royal Society (FRS).

Illness and return to India

Plagued by health problems all his life, in a country far from home, and obsessively involved with his studies, Ramanujan's health worsened in England, perhaps exacerbated by stress, and by the scarcity of vegetarian food during the First World War. He was diagnosed with tuberculosis (Henderson, 1996) and a severe vitamin deficiency, though a 1994 analysis of Ramanujan's medical records and symptoms by Dr. D.A.B Young concluded that it was much more likely he had hepatic amoebiasis, a parasitic infection of the liver. This is also supported by the fact that Ramanujan had spent time in Madras, a coastal city where the disease was widespread. It was a difficult disease to diagnose, but once diagnosed was readily curable (Berndt, 1998). He returned to India in 1919 and died soon after in Kumbakonam, his final gift to the world being the discovery of 'mock Theta functions'. His wife S. Janaki Ammal lived outside Chennai (formerly Madras) until her death in 1994. Janaki had been nine when they were married, a fairly common practice in India at the time. (Henderson, 1996)

Spiritual life

Ramanujan lived as a Tamil Brahmin all his life. Views of his actual beliefs vary: his first Indian biographers described him as rigorously orthodox, whereas G. H. Hardy (an atheist) believed him to be essentially agnostic as far as metaphysical matters were concerned. It could be that like most modern scientists who believe it 'professionally necessary and helpful' to dub themselves atheist, Hardy simply preferred to describe him thus too. At any rate Hinduism itself being an all-inclusive religion - transcending several schools such as atheism, agnosticism, theism, downright materialism, anthropism, etc. - which defies easy comprehension and interpretation even for some interested 'top' scientists, it is not quite meaningful to disagree, either, with Hardy on this description. On the other hand, it is said that Ramanujan, who struggled for a long time with severe illness which tended to impede his mathematical outpouring, said in frustrated agony, while in his death throes, that he did not believe in God.

Hardy reported a statement of Ramanujan's to the effect that all religions are equally correct. Kanigel's biography states that Ramanujan would probably not have shown Hardy

Ramanujan credited his acumen to his family Goddess, Namagiri, and looked to her for inspiration in his work. He often said, "An equation for me has no meaning, unless it represents a thought of God."

Mathematical achievements

In mathematics, there is a distinction between having an insight and having a proof. Ramanujan's talent suggested a plethora of formulae that could then be investigated in depth later. As a byproduct, new directions of research were opened up. Examples of these formulae were intriguing infinite series for π, one of which is given by,

$\backslash frac1\backslash pi\; =\; \backslash frac2\backslash sqrt29801\; \backslash sum^\backslash infty\_k=0\; \backslash frac(4k)!(1103+26390k)(k!)^4\; 396^4k$

which is related to the fact that,

$e^\backslash pi\; \backslash sqrt163\; =\; 640320^3\; -\; 743.99999999999925..$

His intuition had led him to derive some unknown identities. One example is

$\backslash left\; ^-2\; +\; \backslash left\; ^-2\; =\; \backslash frac\; 2\; \backslash Gamma^4\; \backslash left\; (\; \backslash frac34\; \backslash right\; )\backslash pi$

for all $\backslash theta$, where $\backslash Gamma(z)$ is the gamma function. Equating coefficients of $\backslash theta^0$, $\backslash theta^4$, and $\backslash theta^8$ gives some amazing identities for the hyperbolic secant.

Hardy's Quotes

Hardy wrote of Ramanujan:

"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.."

"In his favourite topics, like infinite series and continued fractions, he had no equal this century. His insight into algebraic formulae, often (and unusually) brought about by considering numerical examples, was truly amazing. But in analytic number theory, a subject he is often associated with, I do not believe he actually knew that much. He certainly contributed little of significance that was not known already. And in a subject that relied so much on proof, a subject where intuition had a bad habit of coming unstuck, he produced much that was false."

"I remember once going to see when he was lying ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. 'No,' he replied, 'it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways.'" -G.H. Hardy

"As for his place in the world of Mathematics, we quote Bruce C Berndt: ``Paul Erdos has passed on to us Hardy's personal ratings of mathematicians. Suppose that we rate mathematicians on the basis of pure talent on a scale from 0 to 100, Hardy gave himself a score of 25, J.E. Littlewood 30, Hilbert 80 and Ramanujan 100. G.H.Hardy,"

Theorems and discoveries

These include both Ramanujan's own discoveries, and those developed or proven in collaboration with Hardy.

• Properties of highly composite numbers
• The partition function and its asymptotics
• Mock theta functions

He also made major breakthroughs and discoveries in the areas of:

• Gamma functions
• Modular forms
• Ramanujan's continued fractions
• Divergent series
• Hypergeometric series
• Prime number theory

It is said his discoveries were unusually rich; that is, in many of them there was far more than initially met the eye.

The Ramanujan conjecture and its role

Although there are numerous statements that could bear the name Ramanujan conjecture, there is one in particular that was very influential on later work. That Ramanujan conjecture is an assertion on the size of the coefficients of the tau-function, a typical cusp form in the theory of modular forms. It was finally proved as a consequence of the proof of the Weil conjectures some decades later; the reduction step is complicated.

Ramanujan's notebooks

While he was still in India, Ramanujan recorded many results in three notebooks of loose leaf paper. Results were written up, without their derivations. This is probably the origin of the misconception that Ramanujan was unable to prove his results and simply thought the final result up directly. Berndt, in his review of the notebooks and Ramanujan's work felt that Ramanujan most certainly was able to make the proofs of most of his results, but chose not to.

This style of working may have been for several reasons. Since paper was very expensive, Ramanujan would do most of his work and perhaps his proofs on slate, and then transfer just the results to paper. Using a slate was common for mathematics students in India at the time. He was also quite likely to have been influenced by the style of one of the books he had learned much of his advanced mathematics from G. S. Carr's Synopsis of Pure and Applied Mathematics, used by Carr in his tutoring. It summarised several thousand results, stating them without proofs. Finally, it is possible that Ramanujan considered his workings to be for his personal interest alone; and therefore only recorded the results. (Berndt, 1998)

The first notebook was 351 pages with 16 somewhat organized chapters and some unorganized material. The second notebook had 256 pages in 21 chapters and 100 unorganized pages, with the third notebook containing 33 unorganized pages. The results in his notebooks inspired numerous papers by later mathematicians trying to prove what he had found. Hardy himself created papers exploring material from Ramanujan's work as did G. N. Watson, B. M. Wilson, and Bruce Berndt. (Berndt, 1998)

Recognition

Ramanujan's home state of Tamil Nadu celebrates 22nd December (Ramanujan's birthday) as 'State IT Day', memorializing both the man, and his achievements, as a native of Tamil Nadu.

A Prize for young mathematicians from developing countries has been created in the name of Srinivasa Ramanujan by the International Centre for Theoretical Physics (ICTP), in cooperation with IMU, who nominate members of the Prize Committee.

See also

• Ramanujan-Peterssen conjecture
• 1729 (number)
• Landau-Ramanujan constant
• Ramanujan-Soldner constant
• Ramanujan Rolling Shield
• Ramanujan theta function
• Ramanujan graph
• Ramanujan's tau function
• Rogers-Ramanujan identity
• Ramanujan prime

Further reading

• Collected Papers of Srinivasa Ramanujan ISBN 0821820761
• The Man Who Knew Infinity: A Life of the Genius Ramanujan by Robert Kanigel ISBN 0671750615

References

• An overview of Ramanujan's notebooks by Bruce C. Berndt, in Charlemagne and His Heritage: 1200 Years of Civilization and Science in Europe, Volume 2: Mathematical Arts, P. L. Butzer, H. Th. Jongen, and W. Oberschelp, editors, Brepols, Turnhout, 1998, pp. 119-146, ()
• Modern Mathematicians, Harry Henderson, Facts on File Inc., 1996

Cultural references

• He was referred to in the film Good Will Hunting as an example of mathematical genius.
• His biography was also highlighted in the Vernor Vinge book The Peace War
• The character 'Amita Ramanujan' in the CBS TV series Numb3rs (2005-) was named after him (source: ).

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