Srinivasa Ramanujan

For the algebraic geometer see C. P. Ramanujam.

Srinivasa Ramanujan
Srinivāsa Aiyangār Rāmānujan (1887-1920)
Born	December 22, 1887 Erode, Tanjore, Madras, India
Died	April 26, 1920 Chetput, India
Nationality	Indian
Alma mater	University of Cambridge
Known for	Number Theory, Modular functions
Scientific career
Fields	Mathematician
Doctoral advisor	G H Hardy and John Littlewood

Srinivāsa Aiyangār Rāmānujan (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 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 between 1914 and 1918 at the University of Cambridge. However, Ramanujan was truly a self-thought person and never sought any degree from 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 received similar education 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 (ISBN 1-4181-8509-4), 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 of 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 his 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 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, 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 a number 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.

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.

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.

Spiritual life

Ramanujan lived as a Tamil Brahmin all his life. His first Indian biographers described him as rigorously orthodox. 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,

${\frac {1}{\pi }}={\frac {2{\sqrt {2}}}{9801}}\sum _{k=0}^{\infty }{\frac {(4k)!(1103+26390k)}{(k!)^{4}396^{4k}}}$

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,

$e^{\pi {\sqrt {58}}}=396^{4}-104.000000177\dots .$

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

$\left[1+2\sum _{n=1}^{\infty }{\frac {\cos(n\theta )}{\cosh(n\pi )}}\right]^{-2}+\left[1+2\sum _{n=1}^{\infty }{\frac {\cosh(n\theta )}{\cosh(n\pi )}}\right]^{-2}={\frac {2\Gamma ^{4}\left({\frac {3}{4}}\right)}{\pi }}$

for all $\theta$, where $\Gamma (z)$ is the gamma function. Equating coefficients of $\theta ^{0}$, $\theta ^{4}$, and $\theta ^{8}$ 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.

• 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

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 (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.

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 [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.'"

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.”

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 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.

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

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".

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 0-387-22529-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)

External links

• "Film to celebrate mathematics genius". BBC. Retrieved August 24. {{cite web}}: Check date values in: |accessdate= (help); Unknown parameter |accessyear= ignored (|access-date= suggested) (help)
• O'Connor, John J.; Robertson, Edmund F., "Srinivasa Ramanujan", MacTutor History of Mathematics Archive, University of St Andrews
• Weisstein, Eric Wolfgang (ed.). "Ramanujan, Srinivasa (1887-1920)". 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
• Supernaturalminds
• 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
• [8]
• 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