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Srinivasa Ramanujan

Srinivāsa Aiyangār Rāmānujan (Tamil: ஸ்ரீனிவாஸ ஐயங்கார் ராமானுஜன்) (December 22, 1887 – April 26, 1920) was an Indian mathematician who excelled in the heuristic aspects of number theory and insight into modular functions. He also made significant contributions to the development of partition functionss 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 between 1914 and 1918 at the University of Cambridge. Often, his formulas were merely stated, without proof, and were only later proven to be true. His results were highly original and unconventional, and have inspired a large amount of research and many mathematical papers; however, some of his discoveries have been slow to enter the mathematical mainstream. Recently his formulae have started to be applied in the field of crystallography, and other applications in physics. 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, the place of residence of his maternal grandparents. His father hailed from the fertile Kumbakonam-Tanjore district.They lived in Saarangapani Street in a typical south Indian styled house (it is now a museum). His mother is believed to have been well-educated in Indian mathematics and Ramanujan is conjectured by some to have been as well [1]. In 1898, at age 10, he entered the town high school - THSS in Kumbakonam [2], where he may have encountered formal mathematics for the first time. By the age of 11 he had devoured the mathematical knowledge of two lodgers at his home, both students at the Government College, and was lent books on advanced trigonometry written by S. L. Loney, which he mastered by age 13. His biographer reports that by 14 his true genius was beginning to become discernible. Not only did he achieve merit certificates and academic awards throughout his school career, he 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 was showing familiarity with infinite series. His peers at the time commented later, "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. By age 17, he calculated Euler's constant to 15 decimal places. He began to study what he thought was a new class of numbers, but instead he had independently developed and investigated the Bernoulli numbers. At this time in his life, he was quite poor and was often near the point of starvation.

Adulthood in India

After marriage (on July 14, 1909) he began searching for work. With his packet of mathematical calculations, he travelled around the city of Madras (now Chennai) looking for a clerical position. He managed finally to get a job at the Accountant General's Office at Madras. Ramanujan desired to focus completely on mathematics, and was advised by an Englishman to contact scholars 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 might be the case that he was supported by R.Ramachandra Rao, then the Collector of the Nellore District and a distinguished civil servant. Ramachandra Rao, an amateur mathematician himself was the uncle of the well known mathematician, K. Ananda Rao, who went on to become the Principal of the Presidency college.) It was at this point that Sir Ashutosh Mukherjee tried to bolster 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, recognized 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 [theorem] could have been set in the most advanced mathematical examination in the world.” Although Hardy was one of the pre-eminent mathematicians of his day and an expert in several of the fields Ramanujan was writing about, he commented, "many of them defeated me completely; I had never seen anything in the least like them before."

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. However, Ramanujan's mother said that she had a dream in which the family goddess told her not to stand in the way of her son's travel, so he made plans accordingly, although he took pains to keep a proper Brahmin lifestyle as far as he could.

Hardy said of Ramanujan's formulae, some of which he could not initially understand, "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 to the mathematical giants Euler and Jacobi. Ramanujan was later appointed a Fellow of Trinity, and a Fellow of the Royal Society (FRS).

Some would find Ramanujan scribbling equations in his notebook continuously for more than 30 hours and then collapsing to sleep for 20 hours. This type of irregularity in day to-day activities took a heavy toll on his health.[3]

Illness and return to India

Plagued by health problems all of his life, living 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, although 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, 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 is also said that Ramanujan, who struggled for a long time with severe illness which tended to impede his mathematical output, 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 his religious side in any case; however Kanigel paints a generally negative picture of 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,

based on the negative fundamental discriminant d = −4(58) with class number h(d) = 2 (note that 5×7×13×58 = 26390) and is related to the fact that,

Ramanujan's series for π converges extraordinarily rapidly (exponentially) and forms the basis for the fastest algorithms currently used to calculate π.

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

for all , where is the gamma function. Equating coefficients of , , and gives some deep identities for the hyperbolic secant.

Theorems and discoveries

It is said that Ramanujan's discoveries were unusually rich; that is, in many of them there was far more than initially met the eye. The following include both Ramanujan's own discoveries, and those developed or proven in collaboration with Hardy.

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

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 tau function, which has as generating function the discriminant modular form Δ(q) , a typical cusp form in the theory of modular forms. It was finally proved in 1973, as a consequence of Pierre Deligne's proof of the Weil conjectures; the reduction step is complicated.

Ramanujan's notebooks

While he was still in India, Ramanujan recorded many results in four notebooks of loose leaf paper. Results were written up, without their derivations. This is probably the origin of the perception 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 from which he had learned much of his advanced mathematics: 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) A fourth notebook, the so-called "lost notebook, was rediscovered in 1976.

Excerpts from the correspondences

"Dear Sir,
I beg to introduce myself to you as a clerk in the Accounts Department of the Port Trust Office at Madras on a salary of only 20 Pound per annum. I am now about 23 years of age. I have had no university education but I have undergone the ordinary school course. After leaving the school I have been employing the spare time at my disposal to work at Mathematics. I have not trodden through the conventional regular course which is followed in a university course, but I am striking out a new path for myself. I have made a special investigation of divergent series in general and the results I get are termed by the local mathematicians as 'startling'... Being poor, if you are convinced that there is anything of value I would like to have my theorems published. I have not given the actual investigations nor the expressions that I get but I have indicated to the lines on which I proceed. Being inexperienced I would very highly value any advice you give me. Requesting to be excused for the trouble I give you."

"I was exceedingly interested by your letter and by the theorems which you state. You will however understand that, before I can judge properly of the value of what you have done, it is essential that I should see proofs of some of your assertions. Your results seem to me to fall into roughly three classes:
(1) there are a number of results that are already known, or easily deducible from known theorems;
(2) there are results which, so far as I know, are new and interesting, but interesting rather from their curiosity and apparent difficulty than their importance;
(3) there are results which appear to be new and important..."

"... I have found a friend in you who views my labours sympathetically. This is already some encouragement to me to proceed... I find in many a place in your letter rigorous proofs are required and you ask me to communicate the methods of proof... I told the sum of an infinite number of terms of the series under my theory. If I tell you this you will at once point out to me the lunatic asylum as my goal... What I tell you is this. Verify the results I give and if they agree with you results... you should at least grant that there may be some truths in my fundamental basis...To preserve my brains I want food and this is now my first consideration. Any sympathetic letter from you will be helpful to me here to get a scholarship either from the University or from Government..." (The equation cited above is not as nonsensical as it seems at first sight, and is actually a deep observation about the properties of the Riemann zeta function: see Ramanujan summation for more details.)

"... I am a little pained to see what you have written... I am not in the least apprehensive of my method being utilized by others. On the contrary my method has been in my possession for the last eight years and I have not found anyone to appreciate the method. As I wrote in my last letter I have found a sympathetic friend in you and I am willing to place unreservedly in your hands what little I have. It was on account of the novelty of the method I have used that I am a little diffident even now to communicate my own way of arriving at the expressions I have already given..."

"I am extremely sorry for not writing you a single letter up to now... I discovered very interesting functions recently which I call 'Mock - functions'. Unlike the 'False - function' (studied partially by Prof.Rogers in his interesting paper) they enter into mathematics as beautifully as the ordinary -functions. I am sending you with this letter some examples..."

Hardy on Ramanujan

G. H. Hardy quotes:

  • "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..."

  • "I still say to myself when I am depressed, and find myself forced to listen to pompous and tiresome people, 'Well, I have done one thing you could never have done, and that is to have collaborated with both Littlewood and Ramanujan on something like equal terms.'"

  • "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 Ramanujan 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.'"

  • "...[T]he greatest mathematicians made their most significant discoveries when they were very young. Galois who died at 20, Abel at 26, and Riemann at 39, had actually made their mark in history. So the real tragedy of Ramanujan was not his early death at the age of 32, but that in his most formative years, he did not receive proper training, and so a significant part of his work was rediscovery..."

  • "Immortality may be a silly word but perhaps a mathematician has the best chance of whatever it may mean."

Quoting K. Srinivasa Rao [4]:

"As for his place in the world of Mathematics, we quote Bruce C Berndt: 'Paul Erdos has passed on to us G. H. 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.'"

Narlikar on Ramanujan

In his book, The Scientific Edge, the noted physicist Jayant Narlikar says 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.” Narlikar says that his work was one of the top ten achievements of 20th century Indian science and “could be considered in the Nobel Prize class.” The Scientific Edge by Jayant V. Narlikar, Penguin Books, 2003, page 127. The work of other 20th century Indian scientists which Narlikar considered to be of Nobel Prize class were those of Chandrasekhara Venkata Raman, Meghnad Saha and Satyendra Nath Bose.

Recognition

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

A stamp picturing Ramanujan was released by the Government of India in 1962—the 75th anniversary of Ramanujan's birth—commemorating his achievements in the field of number theory.

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 the IMU, who nominate members of the Prize Committee.

During the year 1987 Ramanujan Centennial, the printed form of Ramanujan's Lost Notebook by Springer-Narosa was released by the late Prime Minister Rajiv Gandhi, who presented the first copy to Janaki Ammal Ramanujan, the late widow of Ramanujan, and the second copy to Professor Andrews in recognition of his contributions in the field of number theory.


Projected films

  • An international feature film on Ramanujan's life will begin shooting in 2007 in Tamil Nadu state and Cambridge. It is being made by an Indo-British collaboration; it will be co-directed by Stephen Fry and Dev Benegal [5]. In October Alter Ego Productions [6] will present Off-Off Broadway with David Freeman's "First Class Man". The play is centered around Ramanujan and his complex and dysfunctional relationship with GH Hardy, an eminent British mathematician and Cambridge don who wants to bring him to Cambridge.
  • 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.[7]

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: IMDB's trivia for 'Numb3rs').
  • The short story "Gomez", by Cyril Kornbluth, mentions Ramanujan by name as a comparison to its title character, another self-taught mathematical genius.

Note

See also

Further reading

  • Collected Papers of Srinivasa Ramanujan ISBN 0-8218-2076-1
  • Ramanujan: Letters and Commentary ISBN 0-8218-0287-9
  • Ramanujan's Notebooks, Part 1 ISBN 0-387-96110-0
  • Ramanujan's Notebooks, Part 2 ISBN 0-387-96794-X
  • Ramanujan's Notebooks, Part 3 ISBN 0-387-97503-9
  • Ramanujan's Notebooks, Part 4 ISBN 0-387-94109-6
  • Ramanujan's Notebooks, Part 5 ISBN 0-387-94941-0
  • The Man Who Knew Infinity: A Life of the Genius Ramanujan by Robert Kanigel ISBN 0-671-75061-5

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, (22 pg. pdf file)
  • Modern Mathematicians, Harry Henderson, Facts on File Inc., 1996
  • Hardy, G. H. (1940) Ramanujan, Cambridge University Press: London (1940). Ams Chelsea Pub. (November 25, 1999) ISBN 0821820230.

External links

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