Srinivasa Ramanujan

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Srinivasa Ramanujan

Srinivāsa Rāmānujan (1887-1920)
Born	December 22, 1887 Erode, Tamil Nadu, India
Died	April 26, 1920 Chetput, (Chennai), Tamil Nadu, India
Residence	India, UK
Nationality	Indian
Field	Mathematician
Alma mater	University of Cambridge
Academic advisor	G. H. Hardy and J. E. Littlewood
Known for	Landau-Ramanujan constant Ramanujan-Soldner constant Ramanujan theta function Rogers-Ramanujan identity Ramanujan prime Mock theta functions Ramanujan's sum
Religion	Hindu

Srinivasa Iyengar Ramanujan (Srinivāsa Iyengār Rāmānujan)(Tamil: ஸ்ரீனிவாஸ ஐயங்கார் ராமானுஜன்) (December 22, 1887 – April 26, 1920) was an Indian mathematician widely regarded as one of the greatest mathematical minds in recent history[1]. With almost no formal mathematical training, he made profound contributions in the areas such as analysis, partition theory and summation formulas for constants such as π. A child prodigy, Ramanujan was largely self-taught and compiled nearly 4,000 theorems during his short lifetime. Although a small number of these theorems were actually false, most of his statements have now been proven to be correct. His deep intuition and uncanny algebraic manipulative ability enabled him to state highly original and unconventional results which defied formal proof for a long time. Ramanujan's theorems have inspired a vast amount of research [2];however,some of his discoveries have been slow to enter the mathematical mainstream. Recently his formulae have begun to be applied in the field of crystallography and physics. The Ramanujan Journal was launched specifically to publish work "in areas of mathematics influenced by Ramanujan".

Contents 1 Life 1.1 Childhood and early life 1.2 Adulthood in India 1.3 Life in England 1.4 Illness and return to India 1.5 Spiritual life 2 Mathematical achievements 2.1 Theorems and discoveries 2.2 The Ramanujan conjecture and its role 2.3 Ramanujan's notebooks 3 Other mathematicians' views of Ramanujan 4 Recognition 5 Projected films 6 Cultural references 7 References 8 See also 9 Selected Publications by Ramanujan 10 Selected Publications about Ramanujan or his work 11 External links

[edit] Life

[edit] Childhood and early life

Ramanujan was born in 1887^[1] in Erode, Tamil Nadu, India, at the place of residence of his maternal grandparents. His father worked as an accountant and hailed from the fertile Thanjavur district. His mother was a housewife and is believed to have been well educated in Indian mathematics. They lived in Saarangapani Street in a south-Indian-styled house (now a museum) in the town of Kumbakonam. In 1898, at the age of 10, Ramanujan entered the town high school, THSS [3], where he encountered formal mathematics for the first time. By the age of 11 he had devoured the mathematical knowledge of two lodgers at his home who were students at the Government College. He was lent advanced trigonometry written by S.L. Loney (ISBN 1-4181-8509-4) and he completely mastered this book by the age of 13 and was discovering sophisticated theorems of his own. His biographer reports that by 14 his true genius was evident. 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 showing familiarity with infinite series. By the age of 17, he had independently developed and investigated the Bernoulli numbers and had calculated Euler's constant up to 15 decimal places. His peers of the time commented later, "We, including teachers, rarely understood him" and "stood in respectful awe" of him.

Ramanujan received a scholarship to study at Government College in Kumbakonam but 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. He failed again in the next college he joined but continued to pursue independent research in mathematics.^[1]At this point in his life, he was financially poor and was quite often near the point of starvation.

[edit] Adulthood in India

After his marriage (in 1909), to a nine-year old bride as per the customs of India at that time ^[1] he began searching for work. With his collection of mathematical discoveries, he travelled door to door around the city of Madras (now Chennai) looking for a clerical position. He finally managed to find a job at the Accountant General's Office. Ramanujan desired to focus completely on mathematics and was advised by his boss, who was also interested in mathematics, to contact scholars in Cambridge.^[1] 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 Ramachandra Rao, then the collector of the Nellore district and a distinguished civil servant. 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.)

In late 1912 and early 1913, Ramanujan sent letters and samples of his theorems to three Cambridge academics: H. F. Baker, E. W. Hobson, and G. H. Hardy. Only Hardy, 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 most eminent mathematicians of his day and an expert in a number of fields that Ramanujan was writing about, he commented that, "many of them defeated me completely; I had never seen anything in the least like them before."

[edit] Life in England

After some initial skepticism (Kanigel 1991 pp161–3), Hardy replied with comments, requesting proofs for some of the discoveries, and began to make plans to bring Ramanujan to Cambridge. Ramanujan was at first reluctant to travel overseas due to religious reasons but eventually agreed to come to England. He spent about five years in Cambridge collaborating with Hardy and Littlewood and published some of his findings there. Ramanujan was appointed a Fellow of Trinity and a Fellow of the Royal Society, which were the highest honors for mathematicians at that time.

[edit] Illness and return to India

Plagued by health problems all through his life, living in a country far away from home, and obsessively involved with his mathematics, 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 and was confined to a sanitorium. Ramanujan returned to India in 1919 and died soon thereafter at the age of 32 in Kumbakonam. His wife, S. Janaki Ammal, lived outside Chennai(formerly Madras)until her death in 1994[4].

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 supported by the fact that Ramanujan had spent time in Madras, where the disease was widespread. He had had two cases of dysentery before he left India. When not properly treated, dysentery can lie dormant for years and lead to hepatic amoebiasis.^[1] It was a difficult disease to diagnose, but once diagnosed would have been readily curable (Berndt, 1998).

[edit] Spiritual life

Ramanujan lived as an observant Tamil Brahmin all his life. The "Iyengar" in his name refers to a subcaste of Brahmins in southern India who are followers of the Hindu god Vishnu, the preserver of the universe. His first Indian biographers described him as rigorously orthodox. Ramanujan credited his acumen to his family goddess, Namagiri, and looked to her for inspiration in his work. He often said, "An equation has no meaning for me, unless it represents a thought of God."

[edit] 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 by-product, new directions of research were opened up. Examples of these formulae include the 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 of the fastest algorithms currently used to calculate π.

His intuition also led him to derive some previously unknown identities, such as

for all θ, where Γ(z) is the gamma function. Equating coefficients of θ⁰, θ⁴, and θ⁸ gives some deep identities for the hyperbolic secant.

[edit] Theorems and discoveries

It is said that Ramanujan's discoveries were unusually rich; that is, in many of them there was far more than what initially met the eye. The following 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

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. A type of prime numbers based on a 1919 publication by Ramanujan is named Ramanujan primes.
• Mock theta functions

[edit] 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 involved is complicated. Deligne won a Fields Medal for his work on Weil conjectures.

[edit] Ramanujan's notebooks

While he was still in India, Ramanujan recorded the bulk of his results in four notebooks of loose leaf paper. Results were mostly written up without any derivations. This is probably the origin of the perception that Ramanujan was unable to prove his results and simply thought up the final result directly. Mathematician Bruce Berndt, in his review of these 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 (ISBN 0-8284-0239-6), 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.

[edit] Other mathematicians' views of 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.'"

• "...[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..."

Quoting K. Srinivasa Rao [5]:

"As for his place in the world of Mathematics, we quote Bruce C Berndt: 'Paul Erdős 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.'"

In his book The Scientific Edge, 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 20^th century Indian science and “could be considered in the Nobel Prize class.” (Narlikar 2003 p127) The work of other 20^th century Indian scientists which Narlikar considered to be of Nobel Prize class were those of Chandrasekhara Venkata Raman, Meghnad Saha and Satyendra Nath Bose.

[edit] 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 75^th 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's 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 S. Janaki Ammal Ramanujan (Ramanujan's late widow) and the second copy to George Andrews in recognition of his contributions in the field of number theory.

[edit] Projected films

• An international feature film on Ramanujan's life will begin shooting in 2007 in Tamil Nadu state and Cambridge. It is being produced by an Indo-British collaboration; it will be co-directed by Stephen Fry and Dev Benegal [6]. In October, Alter Ego Productions [7] 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 G. H. Hardy.
• 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.[8]

[edit] Cultural references

• He was referred to in the film Good Will Hunting as an example of mathematical genius.
• His biography was highlighted in the Vernor Vinge book The Peace War as well as Douglas Hofstadter's Gödel, Escher, Bach.
• 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.
• In the novel Earth by David Brin, the character Jen Wolling uses a representation of Sri Ramanujan as her computer interface.
• In the novel The Peace War by Vernor Vinge, a young mathematical genius is referred to as "my little Ramanujan" accidentally. Then it is hoped the young man doesn't get the connection because, like Ramanujan, the boy is doomed to die prematurely.
• The character "Yugo Amaryl" in Isaac Asimov's Prelude to Foundation is based on Ramanujan.
• The theatre company Complicite have created a production based around the life of Ramanjuan called A Disappearing Number - conceived and directed by Simon McBurney

[edit] References

• Kanigel, Robert. The Man Who Knew Infinity: A Life of the Genius Ramanujan (Washington Square Press, 1991) ISBN 0-671-75061-5
• Narlikar, Jayant V. The Scientific Edge (Penguin Books, 2003)
• Peterson, Doug. "Raiders of the Lost Notebook: LAS mathematician tracks proof for legendary numbers genius.", University of Illinois LASNews, Fall/Winter 2006.

1. ^ ^{a b c d e} Raiders of the Lost Notebook: LAS mathematician tracks proof for legendary numbers genius.. Retrieved on March 20, 2007.

[edit] See also

• List of topics named after Srinivasa Ramanujan
• Ramanujan-Peterssen conjecture
• 1729 (number)
• Landau-Ramanujan constant
• Ramanujan-Soldner constant
• Ramanujan summation
• Ramanujan theta function
• Ramanujan graph
• Ramanujan's tau function
• Rogers-Ramanujan identity
• Ramanujan prime
• Ramanujan's constant (hoax)
• Ramanujan's sum
• Ramanujan modular functions

[edit] Selected Publications by Ramanujan

• Collected Papers of Srinivasa Ramanujan, by Srinivasa Ramanujan, G. H. Hardy, P. V. Seshu Aiyar, B. M. Wilson, Bruce C. Berndt (AMS, 2000, ISBN 0-8218-2076-1)

This book was originally published in 1927 after Ramanujan's death. It contains the 37 papers published in professional journals by Ramanujan during his lifetime. The third re-print contains additional commentary by Bruce C. Berndt.

• Notebooks (2 Volumes), S. Ramanujan, Tata Institute of Fundamental Research, Bombay, 1957.

These books contain photo copies of the original notebooks as written by Ramanujan.

• The Lost Notebook and Other Unpublished Papers, by S. Ramanujan, Narosa, New Delhi, 1988.

This book contains photo copies of the pages in the "Lost Notebook".

[edit] Selected Publications about Ramanujan or his work

• Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work by G. H. Hardy (AMS, 1999, ISBN 0-8218-2023-0)
• Ramanujan: Letters and Commentary (History of Mathematics, V. 9), by Bruce C. Berndt and Robert A. Rankin (American Mathematical Society, 1995, ISBN 0-8218-0287-9)
• Ramanujan: Essays and Surveys (History of Mathematics, V. 22), by Bruce C. Berndt and Robert A. Rankin (American Mathematical Society, 2001, ISBN 0-8218-2624-7)
• Ramanujan's Notebooks, Part I, by Bruce C. Berndt (Springer, 1985, ISBN 0-387-96110-0)
• Ramanujan's Notebooks, Part II, by Bruce C. Berndt (Springer, 1999, ISBN 0-387-96794-X)
• Ramanujan's Notebooks, Part III, by Bruce C. Berndt (Springer, 2004, ISBN 0-387-97503-9)
• Ramanujan's Notebooks, Part IV, by Bruce C. Berndt (Springer, 1993, ISBN 0-387-94109-6)
• Ramanujan's Notebooks, Part V, by Bruce C. Berndt (Springer, 2005, ISBN 0-387-94941-0)
• Ramanujan's Lost Notebook, Part I, by George Andrews and Bruce C. Berndt (Springer, 2005, ISBN 0-387-25529-X)
• Number Theory in the Spirit of Ramanujan by Bruce C. Berndt (AMS, 2006, ISBN 0-8218-4178-5)
• 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, ISBN 0-8160-3235-1)
• The Man Who Knew Infinity: A Life of the Genius Ramanujan by Robert Kanigel (1991, ISBN 0-671-75061-5)

[edit] External links

• Film to celebrate mathematics genius. BBC. Retrieved on August 24, 2006.
• O'Connor, John J., and Edmund F. Robertson. "Srinivasa Ramanujan". MacTutor History of Mathematics archive.
• Eric W. Weisstein, Ramanujan, Srinivasa (1887-1920) at ScienceWorld.
• Biography of this mathematical genius at World of Biography
• A Study Group For Mathematics: Srinivasa Ramanujan Iyengar
• Srinivasan Ramanujan in One Hundred Tamils of 20th Century
• The Ramanujan Journal - An international journal devoted to Ramanujan
• Jai Maharaj, Computing the Mathematical Face of God: S. Ramanujan
• Srinivasa Aiyangar Ramanujan
• A short biography of Ramanujan
• International Math Union Prizes, including a Ramanujan Prize.
• A biographical song about Ramanujan's life
• Feature Film on Math Genius Ramanujan by Dev Benegal and Stephen Fry
• A magical genius
• Norwegian and Indian mathematical geniuses
• RAMANUJAN — Essays and Surveys
• Ramanujan's growing influence
• Ramanujan's mentor
• A biography of Ramanujan set to music
• "A passion for numbers"
• BBC radio programme about Ramanujan - episode 5
• Bruce C. Berndt, Robert A. Rankin. The Books Studied by Ramanujan in India American Mathematical Monthly, Vol. 107, No. 7 (Aug. - Sep., 2000), pp. 595-601 doi:10.2307/2589114
• "Ramanujan's mock theta function puzzle solved"

Persondata
NAME	Ramanujan, Srinivasa
ALTERNATIVE NAMES
SHORT DESCRIPTION	Mathematician
DATE OF BIRTH	December 22, 1887
PLACE OF BIRTH	Erode, Tamil Nadu, India
DATE OF DEATH	April 26, 1920
PLACE OF DEATH	Chetput, (Chennai), Tamil Nadu, India