Srinivasa Ramanujan

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Srinivasa Ramanujan

Born 22 December 1887(1887-12-22)
Erode, Tamil Nadu, India
Died 26 April 1920 (aged 32)
Chetput, (Madras), Tamil Nadu, India
Residence Tamil Nadu, India
Fields Mathematician
Alma mater Trinity College, Cambridge
Academic advisors G. H. Hardy and J. E. Littlewood
Known for Landau-Ramanujan constant

Mock theta functions
Ramanujan prime
Ramanujan-Soldner constant
Ramanujan theta function
Ramanujan's sum

Rogers-Ramanujan identities
Religious stance Hindu

Srīnivāsa Rāmānujan Iyengar FRS, better known as Srinivasa Ramanujan (Tamil: ஸ்ரீநிவாச ராமானுஜன்) (22 December 1887 – 26 April 1920) was an Indian mathematician,[1] who, with almost no formal training in pure mathematics, made substantial contributions to mathematical analysis, number theory, infinite series and continued fractions.

Born and raised in Erode, Tamil Nadu, India, Ramanujan first encountered formal mathematics at age 10. He demonstrated a natural ability, and was given books on advanced trigonometry by S L Loney.[2] He had mastered them by age 13, and even discovered theorems of his own. He demonstrated unusual mathematical skills at school, winning accolades and awards. By 17, Ramanujan conducted his own mathematical research on Bernoulli numbers and the Euler–Mascheroni constant. He received a scholarship to study at Government College in Kumbakonam, but lost it when he failed his non-mathematical coursework. He joined another college to pursue independent mathematical research, working as a clerk in the Accountant-General's office at the Madras Port Trust Office to support himself.[1] In 1912-1913, he sent samples of his theorems to three academicians at the University of Cambridge. Only G. H. Hardy recognized the brilliance of his work, subsequently inviting Ramanujan to study under him at Cambridge.

During his short lifetime, Ramanujan independently compiled nearly 3900 results (mostly identities and equations).[3] Although a small number of these results were actually false and some were already known, most of his claims have now been proven correct.[4] 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.[5] However, some of his major discoveries have been rather slow to enter the mathematical mainstream. Recently, Ramanujan's formulae have found applications in crystallography and string theory. The Ramanujan Journal, an international publication, was launched to publish work in all areas of mathematics influenced by his work.[6]

Contents

[edit] Early life

Ramanujan's home on Sarangapani Street, Kumbakonam.

Ramanujan was born on 22 December 1887 in Erode, Tamil Nadu, India, at the place of residence of his maternal grandparents.[7] His father, K. Srinivasa Iyengar worked as a clerk in a sari shop and hailed from the district of Thanjavur.[8] His mother, Komalatammal or Komal Ammal (Ammal in Tamil is equivalent of Madam in English, Madame in French or Frau in German) was a housewife and also a singer at a local temple. They lived in Sarangapani Street in a south-Indian-style home in the town of Kumbakonam. The family home is now a museum. When Ramanujan was a year and a half old, his mother gave birth to a son named Sadagopan. The newborn died less than three months later. In December 1889, Ramanujan had smallpox and fortunately recovered, unlike the thousands in the Thanjavur district who had succumbed to the disease that year.[9] He moved with his mother to her parents' house in Kanchipuram, near Madras. In November 1891, and again in 1894, his mother gave birth, but both children died before their first birthdays.

On 1 October 1892, Ramanujan was enrolled at the local school.[10] In March 1894, he was moved to a Telugu medium school. After his maternal grandfather lost his job as a court official in Kanchipuram,[11] Ramanujan and his mother moved back to Kumbakonam and he was enrolled in the Kangayan Primary School.[12] After his paternal grandfather died, he was sent back to his maternal grandparents, who were now living in Madras. He did not like school in Madras, and he tried to avoid going to school. His family enlisted a local constable to make sure he would stay in school. Within six months, Ramanujan was back in Kumbakonam again.[12]

Since Ramanujan's father was at work most of the day, his mother took care of him as a child. He had a close relationship with her. From her, he learned about tradition and puranas. He learned to sing religious songs, to attend pujas at the temple and particular eating habits – all of which were necessary for Ramanujan to be a good Brahmin child.[13] At the Kangayan Primary School, Ramanujan performed well. Just before the age of 10, in November 1897, he passed his primary examinations in English, Tamil, geography and arithmetic. With his scores, he finished first in the district.[14] In 1898, his mother gave birth to a healthy boy named Lakshmi Narasimhan.[9] That year, Ramanujan entered Town Higher Secondary School where he encountered formal mathematics for the first time.[14]

By age 11, he had exhausted the mathematical knowledge of two college students, who were lodgers at his home. He was later lent a book on advanced trigonometry written by S.L. Loney.[15][16] He completely mastered this book by the age of 13 and he discovered sophisticated theorems on his own. By 14 he was achieving merit certificates and academic awards which continued throughout his school career and also assisted the school in the logistics of assigning its 1200 students (each with their own needs) to its 35-odd teachers.[17] He completed mathematical exams in half the allotted time, and showed a familiarity with infinite series. When he was 16, Ramanujan came across the book, A Synopsis of Elementary Results in Pure and Applied Mathematics written by George S. Carr.[18] This book was a collection of 5000 theorems, and it introduced Ramanujan to the world of mathematics. The next year, he had independently developed and investigated the Bernoulli numbers and had calculated Euler's constant up to 15 decimal places.[19] His peers of the time commented that they "rarely understood him" and "stood in respectful awe" of him.[17]

When he graduated from Town High in 1904, Ramanujan was awarded the K. Ranganatha Rao prize for mathematics by the school's headmaster, Krishnaswami Iyer. Iyer introduced Ramanujan as an outstanding student who deserved scores higher than the maximum possible marks.[17] He received a scholarship to study at Government College in Kumbakonam,[20] known as the "Cambridge of South India."[21] However, Ramanujan 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.[22] In August 1905, he ran away from home, heading towards Visakhapatnam.[23] He later enrolled at Pachaiyappa's College in Madras. He again excelled in mathematics, but performed poorly in other subjects such as physiology. Ramanujan failed his Fine Arts degree exam in December 1906 and again a year later. 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.[24]

[edit] Adulthood in India

On 14 July 1909, Ramanujan was married to a nine-year old bride, Janaki Ammal.[25] It is important to note that in this particular sect that Ramanujan belonged to, marriage was a formal engagement that was actually consummated only after the bride turned 17 or 18, as per the traditional calendar. After the marriage, Ramanujan developed a hydrocele testis, an abnormal swelling of the tunica vaginalis, an internal membrane in the testicle.[26] The condition could be treated with a routine surgical operation that would release the blocked fluid in the scrotal sac. His family did not have the money for the operation, but in January 1910, a doctor volunteered to do the surgery for free.[27] After his successful surgery, Ramanujan searched for a job. He stayed at friends' houses while he was travelling door to door around the city of Madras (now Chennai) looking for a clerical position. To make some money, he tutored some students at Presidency College who were preparing for their F.A. exam.[28] In late 1910, Ramanujan was sick again, possibly as a result of the surgery earlier in the year. He was fearful for his health, and he even told his friend, R. Radakrishna Iyer, to "hand these [my mathematical notebooks] over to Professor Singaravelu Mudaliar [mathematics professor at Pachaiyappa's College] or to the British professor Edward B. Ross, of the Madras Christian College."[29] After Ramanujan recovered and got back his notebooks from Iyer, he took a northbound train from Kumbakonam to Villupuram, a coastal city under French control.[30][31]

[edit] Attention from mathematicians

He met deputy collector V. Ramaswami Iyer who had recently founded the Indian Mathematical Society.[32] Ramanujan, wishing for a job at the revenue department where Iyer worked, showed him his mathematics notebooks. As Iyer later recalled:

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.[33]

Iyer sent Ramanujan, with letters of introduction, to his mathematician friends in Madras.[32] Some of these friends looked at his work and gave him letters of introduction to R. Ramachandra Rao, the district collector for Nellore and the secretary of the Indian Mathematical Society.[34][35][36] Ramachandra Rao was impressed by Ramanujan's work, but doubted that it was actually his own work. Ramanujan mentioned a correspondence he had with Professor Saldhana, a notable Bombay (now Mumbai) mathematician, in which Saldhana expressed a lack of understanding for his work, but concluded that he was not a phony.[37] Ramanujan's friend, C. V. Rajagopalachari, persisted with Ramachandra Rao and tried to quell any doubts over Ramanujan's academic integrity. Rao agreed to give him another chance, and he listened as Ramanujan discussed elliptic integrals, hypergeometric series, and his theory of divergent series which Rao said ultimately "converted" him to believe Ramanujan's mathematical brilliance.[37] Rao asked him what he wanted, and Ramanujan replied that he needed some work and financial support. Rao consented and sent him to Madras. He continued his mathematical research with Rao's financial aid supporting his daily needs. Ramanujan, with the help of Ramaswami Iyer, had his work published in the Journal of Indian Mathematical Society.[38]

One of the first problems he posed in the journal was:

\sqrt{1+2\sqrt{1+3 \sqrt{1+\cdots}}}.

He waited for a solution to be offered in three issues, over six months, but failed to receive any. At the end, Ramanujan supplied the solution to the problem himself. On page 105 of his first notebook, he formulated an equation that could be used to solve the infinitely nested radicals problem.

x+n+a = \sqrt{ax+(n+a)^2 +x\sqrt{a(x+n)+(n+a)^2+(x+n) \sqrt\mathrm{\cdots}}}

Using this equation, the answer to the question posed in the Journal was simply 3.[39] Ramanujan wrote his first formal paper for the Journal on the properties of Bernoulli numbers. One property he discovered was that the denominators (sequence A027642 in OEIS) of the fractions of Bernoulli numbers were always divisible by six. He also devised a method of calculating Bn based on previous Bernoulli numbers. One of these methods went as follows:

It will be observed that if n is even but not equal to zero,
(i) Bn is a fraction and the numerator of {B_n \over n} in its lowest terms is a prime number,
(ii) the denominator of Bn contains each of the factors 2 and 3 once and only once,
(iii) 2^n(2^n-1){b_n \over n} is an integer and 2(2^n-1)B_n\, consequently is an odd integer.

In "Some Properties of Bernoulli's Numbers", Ramanujan gave three proofs, two corollaries and three conjectures in his 17–page paper.[40] 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 lacking in clearness and precision, that the ordinary [mathematical reader], unaccustomed to such intellectual gymnastics, could hardly follow him.[41]

Ramanujan later wrote another paper and also continued to provide problems in the Journal.[42] In early 1912, he got a temporary job in the Madras Accountant General's office, with a 20 rupee a month salary. He kept the job for only a few weeks.[43] Towards the end of his job at the Account General's office, he applied for a job under the Chief Account of the Madras Port Trust. In a letter dated "9 February 1912", Ramanujan wrote:

Sir,
I understand there is a clerkship vacant in your office, and I beg to apply for the same. I have passed the Matriculation Examination and studied up to the F.A. but was prevented from pursuing my studies further owing to several untoward circumstances. I have, however, been devoting all my time to Mathematics and developing the subject. I can say I am quite confident I can do justice to my work if I am appointed to the post. I therefore beg to request that you will be good enough to confer the appointment on me.[44]

Attached to his application was a recommendation from E. W. Middlemast, a mathematics professor at the Presidence College who wrote that Ramanujan was "a young man of quite exceptional capacity in Mathematics."[45] Three weeks after he had applied, on 1 March, Ramanujan learned that he was accepted for a job as a Class III, Grade IV accounting clerk, making 30 rupees per month.[46] At his office, Ramanujan easily and quickly completed the work he was given, so he spent his spare time doing his mathematical research. Ramanujan's boss, Sir Francis Spring, and S. Narayana Iyer, a colleague who was also treasurer of the Indian Mathematical Society, encouraged Ramanujan in his mathematical pursuits.

[edit] Contacting English mathematicians

Spring, Narayana Iyer, Ramachandra Rao and E. W. Middlemast tried to expose Ramanujan's work to British mathematicians. One mathematician, M. J. M. Hill of University College London, commented that Ramanujan's papers were riddled with holes.[47] He said that although Ramanujan had "a taste for mathematics, and some ability", he lacked the educational background and foundation needed so that his work would be accepted by higher-up mathematicians.[48] Although Hill did not offer to take Ramanujan in as a student, he did give thorough and serious professional advice on his work. With the help of friends, Ramanujan drafted letters to leading mathematicians at Cambridge University.[49]

The first two professors, H. F. Baker and E. W. Hobson, returned Ramanujan's papers without any comments.[50] On 16 January 1913, Ramanujan wrote to G. H. Hardy, who had the foresight to quickly recognize Ramanujan's mathematical skills. The nine pages of mathematical wonder seemed like it could hardly have come from an unestablished mathematician. Hardy originally viewed Ramanujan's manuscripts as a possible "fraud."[51] Hardy knew some of Ramanujan's formulas, but others "seemed scarcely possible to believe."[52] One of the theorems Hardy found hard to believe was found on the bottom of page three (valid for 0<a<b+1/2):

\int_0^\infty \cfrac{1+{x}^2/({b+1})^2}{1+{x}^2/({a})^2} \times\cfrac{1+{x}^2/({b+2})^2}{1+{x}^2/({a+1})^2}\times\cdots\;\;dx = \frac{\sqrt \pi}{2} \times\frac{\Gamma(a+\frac{1}{2})\Gamma(b+1)\Gamma(b-a+\frac{1}{2})}{\Gamma(a)\Gamma(b+\frac{1}{2})\Gamma(b-a+1)}.

Hardy was also impressed by some of Ramanujan's other work relating to infinite series:

1 - 5\left(\frac{1}{2}\right)^3 + 9\left(\frac{1\times3}{2\times4}\right)^3 - 13\left(\frac{1\times3\times5}{2\times4\times6}\right)^3 + \cdots = \frac{2}{\pi}
1 + 9\left(\frac{1}{4}\right)^4 + 17\left(\frac{1\times5}{4\times8}\right)^4 + 25\left(\frac{1\times5\times9}{4\times8\times12}\right)^4 + \cdots = \frac{2^\frac{3}{2}}{\pi^\frac{1}{2}\Gamma^2\left(\frac{3}{4}\right)}.

The first result had already been determined by a mathematician named Bauer. The second one was new to Hardy. It was derived from a class of functions called a hypergeometric series which had first been researched by Leonhard Euler and Carl Friedrich Gauss. Compared to Ramanujan's work on integrals, Hardy found these results "much more intriguing."[53] After he saw Ramanujan's theorems on continued fractions on the last page of the manuscripts, Hardy commented that the "[theorems] defeated me completely; I had never seen anything in the least like them before."[54] He figured that Ramanujan's theorems "must be true, because, if they were not true, no one would have the imagination to invent them."[54] Hardy contacted a colleague, J. E. Littlewood, to take a look at the papers. Littlewood was amazed by the mathematical genius of Ramanujan. After discussing the papers with Littlewood, Hardy concluded that the letters were "certainly the most remarkable I have received" and commented that Ramanujan was "a mathematician of the highest quality, a man of altogether exceptional originality and power."[55] One colleague, E.H. Neville, later commented that "not one [theorem] could have been set in the most advanced mathematical examination in the world."[56]

On 8 February 1913, Hardy wrote a letter back to Ramanujan, expressing his interest for his work. Hardy also added that it was "essential that I should see proofs of some of your assertions."[57] Before his letter arrived in Madras during the third week of February, Hardy contacted the Indian Office to set up plans for Ramanujan's trip to Cambridge. Secretary Arthur Davies of the Advisory Committee for Indian Students met with Ramanujan to discuss the overseas trip.[58] In accordance with his Brahmin upbringing, Ramanujan refused to leave his country to "go to a foreign land."[59] Meanwhile, Ramanujan sent a letter packed with theorems to Hardy, writing, "I have found a friend in you who views my labour sympathetically."[60]

To supplement Hardy's endorsement, a former mathematical lecturer at Trinity College, Cambridge, Gilbert Walker, looked at Ramanujan's work and expressed amazement, urging him to spend time at Cambridge.[61] As a result of Walker's endorsement, B. Hanumantha Rao, a mathematics professor at an engineering college, invited Ramanujan's colleague Narayana Iyer to a meeting of the Board of Studies in Mathematics to discuss "what we can do for S. Ramanujan."[62] The board met and agreed to grant Ramanujan a research scholarship of 75 rupees per month for the next two years at the University of Madras.[63] While he was engaged as a research student, Ramanujan continued to submit papers to the Journal of the Indian Mathematical Society. In one paper, Ramanujan anticipated the work of a Polish mathematician who had published his work shortly after.[64] In his quarterly papers, Ramanujan drew up theorems to make definite integrals more easily solvable. Working off Giuliano Frullani's 1821 integral theorem, Ramanujan formulated generalizations that could be made to evaluate formerly unyielding integrals.[65]

Hardy's correspondence with Ramanujan soured after Ramanujan refused to come to England. Hardy enlisted a colleague lecturing in Madras, E. H. Neville, to mentor and bring Ramanujan to England.[66] Neville asked Ramanujan why he was not coming to Cambridge. Ramanujan apparently had now accepted the proposal, as Neville put it, "Ramanujan needed no converting and that his parents' opposition had been withdrawn."[56] Apparently, Ramanujan's friends convinced his mother to accept the journey to Cambridge. Ramanujan was personally convinced by a vivid dream his mother had, in which the family goddess Namagiri commanded her "to stand no longer between her son and the fullfilment of his life's purpose."[56]

[edit] Life in England

Ramanujan went aboard the S.S. Nevasa on 17 March 1914, and at 10 o'clock in the morning, the ship departed from Madras.[67] He arrived in London on 14 April, with E. H. Neville waiting for him with a car. Four days later, Neville took him to his house on Chesterton Road in Cambridge. Ramanujan immediately began his work with Littlewood and Hardy. After six weeks, Ramanujan moved out of Neville's house and took up residence on Whewell's Court, just a five-minutes walk away from Hardy's room.[68] Hardy and Ramanujan began to take a look at Ramanujan's work in his notebooks. Hardy had already received 120 theorems from Ramanujan in the first two letters, but there were many more results and theorems to be found in the notebooks. Hardy saw that some were wrong, some were already discovered and the rest were new breakthroughs.[69] Ramanujan left a deep impression on Hardy and Littlewood. Littlewood commented, "I can believe that he's at least a Carl Gustav Jacob Jacobi",[70] while Hardy said he "can compare him only with [Leonhard] Euler or Jacobi."[71]

Ramanujan spent nearly five years in Cambridge collaborating with Hardy and Littlewood and published a part of his findings there. Hardy and Ramanujan had highly contrasting personalities. Their collaboration was a clash of different cultures, beliefs and working styles. Hardy was an atheist and an apostle of proof and mathematical rigour, whereas, Ramanujan was a deeply religious man and relied very strongly on his intuition. While in England, Hardy tried his best to fill the gaps in Ramanujan's education without interrupting his spell of inspiration.

Ramanujan was awarded a B.A. degree by research (this degree was later renamed PhD) in March 1916 for his work on highly composite numbers which was published as a paper in the Journal of the London Mathematical Society. The paper was over 50 pages with different properties of such numbers proven. Hardy remarked that this was one of the most unusual papers seen in mathematical research at that time and that Ramanujan showed extraordinary ingenuity in handling it. On 6 December 1917, he was elected to the London Mathematical Society. He was the second Indian to become a Fellow of the Royal Society in 1918, the first having been Ardaseer Cursetjee in 1841, and he became one of the youngest Fellows in the entire history of the Royal Society.[72] He was elected "for his investigation in Elliptic functions and the Theory of Numbers." On 13 October 1918, he became the first Indian to be elected a Fellow of Trinity College, Cambridge.[73]

[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 and a severe vitamin deficiency and was confined to a sanatorium. Ramanujan returned to Kumbakonam, India in 1919 and died soon thereafter at the age of 32. His wife, S. Janaki Ammal, lived in Chennai (formerly Madras) until her death in 1994.[74]

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 two episodes 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.[1]

[edit] Personality and spiritual life

Ramanujan has been described as a person with a somewhat shy and quiet disposition, a dignified man with pleasant manners.[75] He lived a rather spartan life while at Cambridge. Ramanujan's first Indian biographers describe him as rigorously orthodox. Ramanujan credited his acumen to his family goddess, Namagiri, and looked to her for inspiration in his work.[76] He often said, "An equation for me has no meaning, unless it represents a thought of God."[77][78]

G. H. Hardy cites Ramanujan as remarking that all religions seemed equally true to him.[79] Hardy further argued that Ramanujan's religiousness had been overstated—in the point of belief, not practice—by his Indian biographers, and romanticised by Westerners. At the same time, he remarked on Ramanujan's strict observance of vegetarianism.

[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. It is said that Ramanujan's discoveries are unusually rich and that there is often more in it than what initially meets the eye. As a by-product, new directions of research were opened up. Examples of the most interesting of these formulae include the intriguing infinite series for π, one of which is given below

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

This result is 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.

Compare to Heegner numbers, which have class number 1 and yield similar formulae. Ramanujan's series for π converges extraordinarily rapidly (exponentially) and forms the basis of some of the fastest algorithms currently used to calculate π. Truncating the sum to the first term also gives the approximation 9801\sqrt{2}/4412 for π, which is correct to six decimal places.

One of his remarkable capabilities was the rapid solution for problems. He was sharing a room with P. C. Mahalanobis who had a problem, "Imagine that you are on a street with houses marked 1 through n. There is a house in between (x) such that the sum of the house numbers to left of it equals the sum of the house numbers to its right. If n is between 50 and 500, what are n and x." This is a bivariate problem with multiple solutions. Ramanujan thought about it and gave the answer with a twist: He gave a continued fraction. The unusual part was that it was the solution to the whole class of problems. Mahalanobis was astounded and asked how he did it. "It is simple. The minute I heard the problem, I knew that the answer was a continued fraction. Which continued fraction, I asked myself. Then the answer came to my mind", Ramanujan replied.

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

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

In 1918, G. H. Hardy and Ramanujan studied the partition function P(n) extensively and gave a very accurate non-convergent asymptotic series that permits exact computation of the number of partitions of an integer. Hans Rademacher, in 1937, was able to refine their formula to find an exact convergent series solution to this problem. Ramanujan and Hardy's work in this area gave rise to a powerful new method for finding asymptotic formulae, called the circle method.[80]

He discovered mock theta functions in the last year of his life. For many years these functions were a mystery, but they are now known to be the holomorphic parts of harmonic weak Maass forms.

[edit] The Ramanujan conjecture

Main article: Ramanujan-Petersson conjecture

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 André Weil in algebraic geometry opened up new areas of research. 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 proven 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 in 1978 for his work on Weil conjectures.[81]

[edit] Ramanujan's notebooks

Further information: Ramanujan's lost notebook

While still in India, Ramanujan recorded the bulk of his results in four notebooks of loose leaf paper. These results were mostly written up without any derivations. This is probably the origin of the misperception 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.

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 G. S. Carr's book, which stated results without proofs. Finally, it is possible that Ramanujan considered his workings to be for his personal interest alone; and therefore only recorded the results.[82]

The first notebook has 351 pages with 16 somewhat organized chapters and some unorganized material. The second notebook has 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.[82] A fourth notebook, the so-called "lost notebook", was rediscovered in 1976 by George Andrews.[1]

[edit] Hardy-Ramanujan Number 1729

The smallest nontrivial number, i.e., the smallest number representable in two ways as a sum of two cubes. It is given by 1729

1729 = 13 + 123 = 93 + 103.

Also, 1729 is the product of 3 prime numbers:

1729 = 13 + 123 = 93 + 103 = 7 * 13 * 19.

The largest known such number (discovered 12/19/2003 5:56:51am by brute force) is:

885623890831 = 75113 + 77303 = 87593 + 59783 = 3943 * 14737 * 15241.

[edit] Other mathematicians' views of Ramanujan

Ramanujan is generally hailed as an all-time great mathematician, like Leonhard Euler, Carl Friedrich Gauss, and Carl Gustav Jacob Jacobi, for his natural mathematical genius.[83] 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...".[84] Hardy went on to claim that his greatest contribution to mathematics was discovering Ramanujan.

Quoting K. Srinivasa Rao,[85] "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, David Hilbert 80 and Ramanujan 100.'"

In his book Scientific Edge, noted physicist Jayant Narlikar 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." Narlikar also goes on to say that his work was one of the top ten achievements of 20th century Indian science and "could be considered in the Nobel Prize class."[86] The work of other 20th century Indian scientists which Narlikar considered to be of Nobel Prize class were those of Chandrasekhara Venkata Raman, Megh Nad Saha and Satyendra Nath Bose.

[edit] Recognition

Ramanujan's home state of Tamil Nadu celebrates 22 December (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 Ramanujan by the International Centre for Theoretical Physics (ICTP), in cooperation with the International Mathematical Union, who nominate members of the prize committee. During the year 1987 (Ramanujan's centennial), the printed form of Ramanujan's Lost Notebook by the Narosa publishing house of Springer-Verlag was released by the late Indian 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.

Bhavna Thakur, a corporate lawyer based in New York has directed a play titled A First Class Man based on his life.

[edit] Projected films

  • An international feature film on Ramanujan's life was announced in 2006 as due to begin shooting in 2007. It was to be shot in Tamil Nadu state and Cambridge and be produced by an Indo-British collaboration and co-directed by Stephen Fry and Dev Benegal.[87] A play First Class Man by Alter Ego Productions[88] was based on 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.[89]

[edit] See also

[edit] References

  1. ^ a b c d e Peterson, Doug. "Raiders of the Lost Notebook". UIUC College of Liberal Arts and Sciences. http://www.las.uiuc.edu/alumni/news/fall2006/06fall_lostnotebook.html. Retrieved on 2007-06-22. 
  2. ^ Berndt, Bruce C. (2001). Ramanujan: Essays and Surveys. Providence, Rhode Island: American Mathematical Society. pp. 9. ISBN 0-8218-2624-7. 
  3. ^ Berndt, Bruce C. (2005). Ramanujan's Notebooks Part V. SpringerLink. pp. 4. ISBN 0-387-94941-0. 
  4. ^ "Rediscovering Ramanujan". Frontline 16 (17): 650. August 1999. http://www.hinduonnet.com/fline/fl1617/16170810.htm. Retrieved on 2007-06-23. 
  5. ^ Ono, Ken (June-July 2006). "Honoring a Gift from Kumbakonam" (PDF). Notices of the American Mathematical Society 53 (6): 650. doi:10.2307/2589114. http://www.ams.org/notices/200606/fea-ono.pdf. Retrieved on 2007-06-23. 
  6. ^ Alladi, Krishnaswami (1998). Analytic and Elementary Number Theory: A Tribute to Mathematical Legend Paul Erdös. Norwell, Massachusetts: Kluwer Academic Publishers. pp. 6. ISBN 0-7923-8273-0. 
  7. ^ Kanigel, Robert (1991). The Man Who Knew Infinity: A Life of the Genius Ramanujan. New York: Charles Scribner's Sons. pp. 11. ISBN 0-684-19259-4. 
  8. ^ Kanigel (1991), p17-18.
  9. ^ a b Kanigel (1991), p12.
  10. ^ Kanigel (1991), p13.
  11. ^ Kanigel (1991), p19.
  12. ^ a b Kanigel (1991), p14.
  13. ^ Kanigel (1991), p20.
  14. ^ a b Kanigel (1991), p25.
  15. ^ Hardy, G. H. (1999). Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work. Providence, Rhode Island: American Mathematical Society. pp. 2. ISBN 0-8218-2023-0. 
  16. ^ Berndt, Bruce C.; Robert A. Rankin (2001). Ramanujan: Essays and Surveys. Providence, Rhode Island: American Mathematical Society. pp. 9. ISBN 0-8218-2624-7. 
  17. ^ a b c Kanigel (1991), p27.
  18. ^ Kanigel (1991), p39.
  19. ^ Kanigel (1991), p90.
  20. ^ Kanigel (1991), p28.
  21. ^ Kanigel (1991), p45.
  22. ^ Kanigel (1991), p47.
  23. ^ Kanigel (1991), p48-49.
  24. ^ Kanigel (1991), p55-56.
  25. ^ Kanigel (1991), p71.
  26. ^ Kanigel (1991), p72.
  27. ^ Ramanujan, Srinivasa (1968). P. K. Srinivasan. ed. Ramanujan Memorial Number: Letters and Reminiscences. Madras: Muthialpet High School. Vol. 1, p100. 
  28. ^ Kanigel (1991), p73.
  29. ^ Kanigel (1991), p74-75.
  30. ^ Ranganathan, S. R. (1967). Ramanujan: The Man and the Mathematician. Bombay: Asia Publishing House. pp. 23. 
  31. ^ Srinivasan (1968), Vol. 1, p99.
  32. ^ a b Kanigel (1991), p77.
  33. ^ Srinivasan (1968), Vol. 1, p129.
  34. ^ Srinivasan (1968), Vol. 1, p86.
  35. ^ Neville, Eric Harold (January 1921). "The Late Srinivasa Ramanujan". Nature 106 (2673): 661–662. doi:10.1038/106661b0. 
  36. ^ Ranganathan (1967), p24.
  37. ^ a b Kanigel (1991), p80.
  38. ^ Kanigel (1991), p86.
  39. ^ Kanigel (1991), p87.
  40. ^ Kanigel (1991), p91.
  41. ^ Seshu Iyer, P. V. (June 1920). "The Late Mr. S. Ramanujan, B.A., F.R.S.". Journal of the Indian Mathematical Society 12 (3): 83. 
  42. ^ Neville (March 1942), p292.
  43. ^ Srinivasan (1968), p176.
  44. ^ Srinivasan (1968), p31.
  45. ^ Srinivasan (1968), p49.
  46. ^ Kanigel (1991), p96.
  47. ^ Kanigel (1991), p105.
  48. ^ Letter from M. J. M. Hill to a C. L. T. Griffith (a former student who sent the request to Hill on Ramanujan's behalf), 28 November 1912.
  49. ^ Kanigel (1991), p106.
  50. ^ Kanigel (1991), p170-171.
  51. ^ Snow, C. P. (1966). Variety of Men. New York: Charles Scribner's Sons. pp. 30–31. 
  52. ^ Hardy, G. H. (June 1920). "Obituary, S. Ramanujan". Nature 105: 494. doi:10.2307/2589114. 
  53. ^ Kanigel (1991), p167.
  54. ^ a b Kanigel (1991), p168.
  55. ^ Hardy (June 1920), p494-495.
  56. ^ a b c Neville, Eric Harold (March 1942). "Srinivasa Ramanujan". Nature 149 (3776): 293. doi:10.1038/149292a0. 
  57. ^ Letter, Hardy to Ramanujan, 8 February 1913.
  58. ^ Letter, Ramanujan to Hardy, 22 January 1914.
  59. ^ Kanigel (1991), p185.
  60. ^ Letter, Ramanujan to Hardy, 27 February 1913, Cambridge University Library.
  61. ^ Kanigel (1991), p175.
  62. ^ Ram, Suresh (1972). Srinivasa Ramanujan. New Delhi: National Book Trust. pp. 29. 
  63. ^ Ranganathan (1967), p30-31.
  64. ^ Ranganathan (1967), p12.
  65. ^ Kanigel (1991), p183.
  66. ^ Kanigel (1991), p184.
  67. ^ Kanigel (1991), p196.
  68. ^ Kanigel (1991), p202.
  69. ^ Hardy, G. H. (1940). Ramanujan. Cambridge: Cambridge University Press. pp. 10. 
  70. ^ Letter, Littlewood to Hardy, early March 1913.
  71. ^ Hardy, G. H. (1979). Collected Papers of G. H. Hardy. Oxford, England: Clarendon Press. Vol. 7, p720. 
  72. ^ Kanigel (1991), p295.
  73. ^ Kanigel (1991), p299-300.
  74. ^ "Ramanujan’s wife: Janakiammal (Janaki)" (PDF). http://www.imsc.res.in/~rao/ramanujan/newnow/janaki.pdf. 
  75. ^ "Ramanujan's Personality". http://www.imsc.res.in/~rao/ramanujan/newnow/pcm5.htm. 
  76. ^ Kanigel (1991), p36.
  77. ^ "Quote by Srinivasa Ramanujan Iyengar". http://lagrange.math.trinity.edu/aholder/misc/quotes.shtml. 
  78. ^ Chaitin, Gregory (2007-07-28). "Less Proof, More Truth". NewScientist 107 (2614): 49. doi:10.2307/2589114. 
  79. ^ Kanigel (1991), p283.
  80. ^ "Partition Formula". http://mathworld.wolfram.com/PartitionFunctionP.html. 
  81. ^ Ono (June-July 2006), p649.
  82. ^ a b "Ramanujans Notebooks". http://www.amazon.com/Ramanujans-Notebooks-Part-Bruce-Berndt/dp/0387949410. 
  83. ^ K. Srinivasa Rao, "Srinivasa Ramanujan". http://www.imsc.res.in/~rao/ramanujan.html. 
  84. ^ "Ramanujan quote". http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Ramanujan.html. 
  85. ^ K Srinivasa Rao. "Srinivasa Ramanujan (22 December 1887 - 26 April 1920)". http://www.imsc.res.in/~rao/ramanujan.html. 
  86. ^ "Narlikar's book". http://www.amazon.com/Scientific-Edge-Indian-Scientist-Modern/dp/0143030280. 
  87. ^ "Film to celebrate maths genius". BBC News. 2006-03-16. http://news.bbc.co.uk/2/hi/south_asia/4811920.stm. Retrieved on 2008-08-08. 
  88. ^ First Class Man
  89. ^ Two Hollywood movies on Ramanujan

[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 and his work

  • Berndt, Bruce C. "An Overview of Ramanujan's Notebooks." Charlemagne and His Heritage: 1200 Years of Civilization and Science in Europe. Ed. P. L. Butzer, W. Oberschelp, and H. Th. Jongen. Turnhout, Belgium: Brepols, 1998. 119-146. Text
  • Berndt, Bruce C., and George E. Andrews. Ramanujan's Lost Notebook, Part I. New York: Springer, 2005. ISBN 0-387-25529-X.
  • Berndt, Bruce C., and George E. Andrews. Ramanujan's Lost Notebook, Part II. New York: Springer, 2008. ISBN 9780387777658
  • Berndt, Bruce C., and Robert A. Rankin. Ramanujan: Letters and Commentary. Vol. 9. Providence, Rhode Island: American Mathematical Society, 1995. ISBN 0-8218-0287-9.
  • Berndt, Bruce C., and Robert A. Rankin. Ramanujan: Essays and Surveys. Vol. 22. Providence, Rhode Island: American Mathematical Society, 2001. ISBN 0-8218-2624-7.
  • Berndt, Bruce C. Number Theory in the Spirit of Ramanujan. Providence, Rhode Island: American Mathematical Society, 2006. ISBN 0-8218-4178-5.
  • Berndt, Bruce C. Ramanujan's Notebooks, Part I. New York: Springer, 1985. ISBN 0-387-96110-0.
  • Berndt, Bruce C. Ramanujan's Notebooks, Part II. New York: Springer, 1999. ISBN 0-387-96794-X.
  • Berndt, Bruce C. Ramanujan's Notebooks, Part III. New York: Springer, 2004. ISBN 0-387-97503-9.
  • Berndt, Bruce C. Ramanujan's Notebooks, Part IV. New York: Springer, 1993. ISBN 0-387-94109-6.
  • Berndt, Bruce C. Ramanujan's Notebooks, Part V. New York: Springer, 2005. ISBN 0-387-94941-0.
  • Hardy, G. H. Ramanujan. New York, Chelsea Pub. Co., 1978. ISBN 0-8284-0136-5
  • Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work. Providence, Rhode Island: American Mathematical Society, 1999. ISBN 0-8218-2023-0.
  • Henderson, Harry. Modern Mathematicians. New York: Facts on File Inc., 1995. ISBN 0-8160-3235-1.
  • Kanigel, Robert. The Man Who Knew Infinity: a Life of the Genius Ramanujan. New York: Charles Scribner's Sons, 1991. ISBN 0-684-19259-4.
  • Leavitt, David. The Indian Clerk. London: Bloomsbury, 2007. ISBN 978-0-7475-9370-6 (paperback).
  • Narlikar, Jayant V. Scientific Edge: the Indian Scientist From Vedic to Modern Times. New Delhi, India: Penguin Books, 2003. ISBN 0143030280.
  • T.M.Sankaran. "Srinivasa Ramanujan- Ganitha lokathile Mahaprathibha", (in Malayalam), 2005, Kerala Sastra Sahithya Parishath, Kochi.

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Persondata
NAME Ramanujan, Srinivasa
ALTERNATIVE NAMES
SHORT DESCRIPTION Mathematician
DATE OF BIRTH 22 December 1887
PLACE OF BIRTH Erode, Tamil Nadu, India
DATE OF DEATH 26 April 1920
PLACE OF DEATH Chetput, (Madras), Tamil Nadu, India
Retrieved from "http://en.wikipedia.org/wiki/Srinivasa_Ramanujan"
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