Ramanujan srinivasa

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Srinivasa RamanujanFrom Wikipedia, the free encyclopediaJump to: navigation, search"Ramanujan" redirects here. For other uses, see Ramanujan (disambiguation).Srinivasa RamanujanBorn 22 December 1887Erode, Madras PresidencyDied 26 April 1920 (aged 32)Chetput, Madras, Madras PresidencyNationality IndiaFields Mathematics

Alma mater Government Arts CollegePachaiyappa's CollegeThe University of Cambridge

Academic advisors G. H. HardyJ. E. LittlewoodKnown for LandauRamanujan constantMock theta functionsRamanujan conjectureRamanujan prime

RamanujanSoldner constantRamanujan theta functionRamanujan's sumRogersRamanujan identitiesRamanujan's master theoremInfluences G. H. Hardy

Srniv sa R m nujan FRS About this sound pronunciation (helpinfo) (Tamil: ) (22 December 1887 26 April 1920) was an Indian mathematician and autodidactwho, with almost no formal training in pure mathematics, made extraordinary contributions tomathematical analysis, number theory, infinite series and continued fractions. Ramanujan was saidby the English mathematician G.H. Hardy to be in the same league as mathematicians like Eulerand Gauss in terms of natural genius.[1]

Born in Erode, Madras Presidency, to a poor Brahmin family, Ramanujan first encountered formalmathematics at age 10. He demonstrated a natural ability, and was given books on advancedtrigonometry written by S. L. Loney.[2] He mastered them by age 12, and even discoveredtheorems of his own, including independently re-discovering Euler's identity. He demonstratedunusual mathematical skills at school, winning accolades and awards. By 17, Ramanujanconducted his own mathematical research on Bernoulli numbers and the EulerMascheroniconstant. He received a scholarship to study at Government College in Kumbakonam, but lost itwhen he failed his non-mathematical coursework. He joined another college to pursue independentmathematical research, working as a clerk in the Accountant-General's office at the Madras PortTrust Office to support himself.[3] In 19121913, he sent samples of his theorems to threeacademics at the University of Cambridge. Only Hardy recognised the brilliance of his work,subsequently inviting Ramanujan to visit and work with him at Cambridge. He became a Fellow of

the Royal Society and a Fellow of Trinity College, Cambridge, dying of illness, malnutrition andpossibly liver infection in 1920 at the age of 32.

During his short lifetime, Ramanujan independently compiled nearly 3900 results (mostly identitiesand equations).[4] Although a small number of these results were actually false and some werealready known, most of his claims have now been proven correct.[5] He stated results that wereboth original and highly unconventional, such as the Ramanujan prime and the Ramanujan thetafunction, and these have inspired a vast amount of further research.[6] However, the mathematicalmainstream has been rather slow in absorbing some of his major discoveries. The RamanujanJournal, an international publication, was launched to publish work in all areas of mathematics


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influenced by his work.[7]

In recognition of his contribution to mathematics, the Government of India declared in Dec 2011 tocelebrate Ramanujan's birthday as 'National Mathematics Day' every year on 22 December anddeclared 2012 as the 'National Mathematical Year'.[8]Contents

1 Early life2 Adulthood in India

2.1 Attention from mathematicians2.2 Contacting English mathematicians

3 Life in England3.1 Illness and return to India3.2 Personality and spiritual life

4 Mathematical achievements4.1 The Ramanujan conjecture4.2 Ramanujan's notebooks

5 RamanujanHardy number 17296 Other mathematicians' views of Ramanujan7 Recognition

8 In popular culture9 See also10 Notes11 Selected publications by Ramanujan12 Selected publications about Ramanujan and his work13 External links

13.1 Media links13.2 Biographical links13.3 Other links

Early lifeRamanujan's home on Sarangapani Street, Kumbakonam.

Ramanujan was born on 22 December 1887 in the city Erode, Madras Presidency, at the residenceof his maternal grandparents.[9] His father, K. Srinivasa Iyengar, worked as a clerk in a sari shopand hailed from the district of Thanjavur.[10] His mother, Komalatammal, was a housewife and alsosang at a local temple.[11] They lived in Sarangapani Street in a traditional home in the town ofKumbakonam. The family home is now a museum. When Ramanujan was a year and a half old, hismother gave birth to a son named Sadagopan, who died less than three months later. In December1889, Ramanujan had smallpox and recovered, unlike thousands in the Thanjavur District who diedfrom the disease that year.[12] He moved with his mother to her parents' house in Kanchipuram,near Madras (now Chennai). In November 1891, and again in 1894, his mother gave birth, but bothchildren died in infancy.

On 1 October 1892, Ramanujan was enrolled at the local school.[13] In March 1894, he was movedto a Telugu medium school. After his maternal grandfather lost his job as a court official in

Kanchipuram,[14] Ramanujan and his mother moved back to Kumbakonam and he was enrolled inthe Kangayan Primary School.[15] After his paternal grandfather died, he was sent back to hismaternal grandparents, who were now living in Madras. He did not like school in Madras, and hetried to avoid attending. His family enlisted a local constable to make sure he attended school.Within six months, Ramanujan was back in Kumbakonam.[15]

Since Ramanujan's father was at work most of the day, his mother took care of him as a child. Hehad a close relationship with her. From her, he learned about tradition and puranas. He learned tosing religious songs, to attend pujas at the temple and particular eating habits all of which are partof Brahmin culture.[16] At the Kangayan Primary School, Ramanujan performed well. Just before


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the age of 10, in November 1897, he passed his primary examinations in English, Tamil, geographyand arithmetic. With his scores, he stood first in the district.[17] That year, Ramanujan enteredTown Higher Secondary School where he encountered formal mathematics for the first time.[17]

By age 11, he had exhausted the mathematical knowledge of two college students who werelodgers at his home. He was later lent a book on advanced trigonometry written by S. L. Loney.[18][19] He completely mastered this book by the age of 13 and discovered sophisticated theorems onhis own. By 14, he was receiving merit certificates and academic awards which continuedthroughout his school career and also assisted the school in the logistics of assigning its 1200students (each with their own needs) to its 35-odd teachers.[20] He completed mathematical examsin half the allotted time, and showed a familiarity with infinite series. In 1903 when he was 16,Ramanujan obtained from a friend a library-loaned copy of a book by G. S. Carr.[21][22] The bookwas titled A Synopsis of Elementary Results in Pure and Applied Mathematics and was a collectionof 5000 theorems. Ramanujan reportedly studied the contents of the book in detail.[23] The book isgenerally acknowledged as a key element in awakening the genius of Ramanujan.[23] The nextyear, he had independently developed and investigated the Bernoulli numbers and had calculatedEuler's constant up to 15 decimal places.[24] His peers at the time commented that they "rarelyunderstood him" and "stood in respectful awe" of him.[20]

When he graduated from Town Higher Secondary School in 1904, Ramanujan was awarded the K.

Ranganatha Rao prize for mathematics by the school's headmaster, Krishnaswami Iyer. Iyerintroduced Ramanujan as an outstanding student who deserved scores higher than the maximumpossible marks.[20] He received a scholarship to study at Government Arts College, Kumbakonam,[25][26] However, Ramanujan was so intent on studying mathematics that he could not focus onany other subjects and failed most of them, losing his scholarship in the process.[27] In August1905, he ran away from home, heading towards Visakhapatnam and stayed in Rajahmundry forabout a month.[28] He later enrolled at Pachaiyappa's College in Madras. He again excelled inmathematics 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 andcontinued to pursue independent research in mathematics. At this point in his life, he lived inextreme poverty and was often on the brink of starvation.[29]

Adulthood in India

On 14 July 1909, Ramanujan was married to a nine-year old bride, Janaki Ammal.[30] After themarriage, Ramanujan developed a hydrocele testis, an abnormal swelling of the tunica vaginalis,an internal membrane in the testicle.[31] The condition could be treated with a routine surgicaloperation that would release the blocked fluid in the scrotal sac. His family did not have the moneyfor the operation, but in January 1910, a doctor volunteered to do the surgery for free.[32]

After his successful surgery, Ramanujan searched for a job. He stayed at friends' houses while hewent door to door around the city of Madras (now Chennai) looking for a clerical position. To makesome money, he tutored some students at Presidency College who were preparing for their F.A.exam.[33]

In late 1910, Ramanujan was sick again, possibly as a result of the surgery earlier in the year. Hefeared for his health, and even told his friend, R. Radakrishna Iyer, to "hand these [my

mathematical notebooks] over to Professor Singaravelu Mudaliar [mathematics professor atPachaiyappa's College] or to the British professor Edward B. Ross, of the Madras ChristianCollege."[34] After Ramanujan recovered and got back his notebooks from Iyer, he took anorthbound train from Kumbakonam to Villupuram, a coastal city under French control.[35][36]

Attention from mathematicians

He met deputy collector V. Ramaswamy Aiyer, who had recently founded the Indian MathematicalSociety.[37] Ramanujan, wishing for a job at the revenue department where Ramaswamy Aiyerworked, showed him his mathematics notebooks. As Ramaswamy Aiyer later recalled:


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I was struck by the extraordinary mathematical results contained in it [the notebooks]. I had nomind to smother his genius by an appointment in the lowest rungs of the revenue department.[38]

Ramaswamy Aiyer sent Ramanujan, with letters of introduction, to his mathematician friends inMadras.[37] 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 MathematicalSociety.[39][40][41] Ramachandra Rao was impressed by Ramanujan's research but doubted that itwas actually his own work. Ramanujan mentioned a correspondence he had with ProfessorSaldhana, a notable Bombay mathematician, in which Saldhana expressed a lack of understandingfor his work but concluded that he was not a phony.[42] Ramanujan's friend, C. V. Rajagopalachari,persisted with Ramachandra Rao and tried to quell any doubts over Ramanujan's academicintegrity. Rao agreed to give him another chance, and he listened as Ramanujan discussed ellipticintegrals, hypergeometric series, and his theory of divergent series, which Rao said ultimately"converted" him to a belief in Ramanujan's mathematical brilliance.[42] When Rao asked him whathe wanted, Ramanujan replied that he needed some work and financial support. Rao consentedand sent him to Madras. He continued his mathematical research with Rao's financial aid takingcare of his daily needs. Ramanujan, with the help of Ramaswamy Aiyer, had his work published inthe Journal of Indian Mathematical Society.[43]

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. Atthe end, Ramanujan supplied the solution to the problem himself. On page 105 of his firstnotebook, he formulated an equation that could be used to solve the infinitely nested radicalsproblem.

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

Using this equation, the answer to the question posed in the Journal was simply 3.[44] Ramanujanwrote his first formal paper for the Journal on the properties of Bernoulli numbers. One property hediscovered 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 onprevious 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 his 17-page paper, "Some Properties of Bernoulli's Numbers", Ramanujan gave three proofs, twocorollaries and three conjectures.[45] Ramanujan's writing initially had many flaws. As Journaleditor 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 intellectualgymnastics, could hardly follow him.[46]

Ramanujan later wrote another paper and also continued to provide problems in the Journal.[47] Inearly 1912, he got a temporary job in the Madras Accountant General's office, with a salary of 20rupees per month. He lasted for only a few weeks.[48] Toward the end of that assignment heapplied for a position under the Chief Accountant of the Madras Port Trust. In a letter dated 9February 1912, Ramanujan wrote:

Sir,


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I understand there is a clerkship vacant in your office, and I beg to apply for the same. I havepassed the Matriculation Examination and studied up to the F.A. but was prevented from pursuingmy studies further owing to several untoward circumstances. I have, however, been devoting all mytime to Mathematics and developing the subject. I can say I am quite confident I can do justice tomy work if I am appointed to the post. I therefore beg to request that you will be good enough toconfer the appointment on me.[49]

Attached to his application was a recommendation from E. W. Middlemast, a mathematicsprofessor at the Presidency College, who wrote that Ramanujan was "a young man of quiteexceptional capacity in Mathematics".[50] Three weeks after he had applied, on 1 March,Ramanujan learned that he had been accepted as a Class III, Grade IV accounting clerk, making30 rupees per month.[51] At his office, Ramanujan easily and quickly completed the work he wasgiven, so he spent his spare time doing mathematical research. Ramanujan's boss, Sir FrancisSpring, and S. Narayana Iyer, a colleague who was also treasurer of the Indian MathematicalSociety, encouraged Ramanujan in his mathematical pursuits.Contacting English mathematicians

On the spring of 1913, Narayana Iyer, Ramachandra Rao and E. W. Middlemast tried to presentRamanujan's work to British mathematicians. One mathematician, M. J. M. Hill of UniversityCollege London, commented that Ramanujan's papers were riddled with holes.[52] He said that

although Ramanujan had "a taste for mathematics, and some ability", he lacked the educationalbackground and foundation needed to be accepted by mathematicians.[53] Although Hill did notoffer to take Ramanujan on as a student, he did give thorough and serious professional advice onhis work. With the help of friends, Ramanujan drafted letters to leading mathematicians atCambridge University.[54]

The first two professors, H. F. Baker and E. W. Hobson, returned Ramanujan's papers withoutcomment.[55] On 16 January 1913, Ramanujan wrote to G. H. Hardy. Coming from an unknownmathematician, the nine pages of mathematical wonder made Hardy initially view Ramanujan'smanuscripts as a possible "fraud".[56] Hardy recognised some of Ramanujan's formulae but others"seemed scarcely possible to believe".[57] One of the theorems Hardy found so incredible wasfound 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 onewas new to Hardy, and was derived from a class of functions called a hypergeometric series whichhad first been researched by Leonhard Euler and Carl Friedrich Gauss. Compared to Ramanujan'swork on integrals, Hardy found these results "much more intriguing".[58] After he saw Ramanujan'stheorems 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".[59]He figured that Ramanujan's theorems "must be true, because, if they were not true, no one wouldhave the imagination to invent them".[59] Hardy asked a colleague, J. E. Littlewood, to take a lookat the papers. Littlewood was amazed by the mathematical genius of Ramanujan. After discussingthe papers with Littlewood, Hardy concluded that the letters were "certainly the most remarkable I


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have received" and commented that Ramanujan was "a mathematician of the highest quality, aman of altogether exceptional originality and power".[60] One colleague, E. H. Neville, latercommented that "not one [theorem] could have been set in the most advanced mathematicalexamination in the world".[61]

On 8 February 1913, Hardy wrote a letter to Ramanujan, expressing his interest for his work. Hardyalso added that it was "essential that I should see proofs of some of your assertions".[62] Before hisletter arrived in Madras during the third week of February, Hardy contacted the Indian Office to planfor Ramanujan's trip to Cambridge. Secretary Arthur Davies of the Advisory Committee for IndianStudents met with Ramanujan to discuss the overseas trip.[63] In accordance with his Brahminupbringing, Ramanujan refused to leave his country to "go to a foreign land".[64] Meanwhile,Ramanujan sent a letter packed with theorems to Hardy, writing, "I have found a friend in you whoviews my labour sympathetically."[65]

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 timeat Cambridge.[66] As a result of Walker's endorsement, B. Hanumantha Rao, a mathematicsprofessor at an engineering college, invited Ramanujan's colleague Narayana Iyer to a meeting ofthe Board of Studies in Mathematics to discuss "what we can do for S. Ramanujan".[67] The boardagreed to grant Ramanujan a research scholarship of 75 rupees per month for the next two years

at the University of Madras.[68] While he was engaged as a research student, Ramanujancontinued to submit papers to the Journal of the Indian Mathematical Society. In one instance,Narayana Iyer submitted some theorems of Ramanujan on summation of series to the abovemathematical journal adding The following theorem is due to S. Ramanujan, the mathematicsstudent of Madras University. Later in November, British Professor Edward B. Ross of MadrasChristian College, whom Ramanujan had met a few years before, stormed into his class one daywith his eyes glowing, asking his students, Does Ramanujan know Polish? The reason was that inone paper, Ramanujan had anticipated the work of a Polish mathematician whose paper had justarrived by the days mail.[69] In his quarterly papers, Ramanujan drew up theorems to makedefinite integrals more easily solvable. Working off Giuliano Frullani's 1821 integral theorem,Ramanujan formulated generalisations that could be made to evaluate formerly unyieldingintegrals.[70]

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 toEngland.[71] Neville asked Ramanujan why he would not go to Cambridge. Ramanujan apparentlyhad now accepted the proposal; as Neville put it, "Ramanujan needed no converting and that hisparents' opposition had been withdrawn".[61] Apparently, Ramanujan's mother had a vivid dream inwhich the family Goddess Namagiri commanded her "to stand no longer between her son and thefulfilment of his life's purpose".[61]Life in EnglandRamanujan (centre) with other scientists at Trinity CollegeWhewell's Court, Cambridge

Ramanujan boarded the S.S. Nevasa on 17 March 1914, and at 10 o'clock in the morning, the shipdeparted from Madras.[72] 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. Ramanujanimmediately began his work with Littlewood and Hardy. After six weeks, Ramanujan moved out ofNeville's house and took up residence on Whewell's Court, just a five-minute walk from Hardy'sroom.[73] Hardy and Ramanujan began to take a look at Ramanujan's notebooks. Hardy hadalready received 120 theorems from Ramanujan in the first two letters, but there were many moreresults and theorems to be found in the notebooks. Hardy saw that some were wrong, others hadalready been discovered, while the rest were new breakthroughs.[74] Ramanujan left a deepimpression on Hardy and Littlewood. Littlewood commented, "I can believe that he's at least aJacobi",[75] while Hardy said he "can compare him only with [Leonhard] Euler or Jacobi."[76]


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Ramanujan spent nearly five years in Cambridge collaborating with Hardy and Littlewood andpublished 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 anatheist and an apostle of proof and mathematical rigour, whereas Ramanujan was a deeplyreligious man and relied very strongly on his intuition. While in England, Hardy tried his best to fillthe 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 March1916 for his work on highly composite numbers, which was published as a paper in the Journal ofthe London Mathematical Society. The paper was over 50 pages with different properties of suchnumbers proven. Hardy remarked that this was one of the most unusual papers seen inmathematical research at that time and that Ramanujan showed extraordinary ingenuity in handlingit. On 6 December 1917, he was elected to the London Mathematical Society. He became a Fellowof the Royal Society in 1918, becoming the second Indian to do so, following Ardaseer Cursetjee in1841, and he was one of the youngest Fellows in the history of the Royal Society. He was elected"for his investigation in Elliptic functions and the Theory of Numbers." On 13 October 1918, hebecame the first Indian to be elected a Fellow of Trinity College, Cambridge.[77]Illness and return to India

Plagued by health problems throughout his life, living in a country far away from home, and

obsessively involved with his mathematics, Ramanujan's health worsened in England, perhapsexacerbated by stress and by the scarcity of vegetarian food during the First World War. He wasdiagnosed with tuberculosis and a severe vitamin deficiency and was confined to a sanatorium.

Ramanujan returned to Kumbakonam, Madras Presidency in 1919 and died soon thereafter at theage of 32. His widow, S. Janaki Ammal, lived in Chennai (formerly Madras) until her death in 1994.[78]

A 1994 analysis of Ramanujan's medical records and symptoms by Dr. D.A.B. Young concludedthat it was much more likely he had hepatic amoebiasis, a parasitic infection of the liver widespreadin Madras, where Ramanujan had spent time. He had two episodes of dysentery before he leftIndia. When not properly treated dysentery can lie dormant for years and lead to hepaticamoebiasis,[3] a difficult disease to diagnose, but once diagnosed readily cured.[3]

Personality and spiritual life

Ramanujan has been described as a person with a somewhat shy and quiet disposition, a dignifiedman with pleasant manners.[79] He lived a rather Spartan life while at Cambridge. Ramanujan'sfirst Indian biographers describe him as rigorously orthodox. Ramanujan credited his acumen to hisfamily Goddess, Namagiri of Namakkal. He looked to her for inspiration in his work,[80] and claimedto dream of blood drops that symbolised her male consort, Narasimha, after which he wouldreceive visions of scrolls of complex mathematical content unfolding before his eyes.[81] He oftensaid, "An equation for me has no meaning, unless it represents a thought of God."[82][83]

Hardy cites Ramanujan as remarking that all religions seemed equally true to him.[84] Hardy furtherargued that Ramanujan's religiousness had been romanticised by Westerners and overstatedinreference to his belief, not practiceby Indian biographers. At the same time, he remarked on

Ramanujan's strict observance of vegetarianism.Mathematical achievements

In mathematics, there is a distinction between having an insight and having a proof. Ramanujan'stalent suggested a plethora of formulae that could then be investigated in depth later. It is said thatRamanujan's discoveries are unusually rich and that there is often more to them than initially meetsthe eye. As a by-product, new directions of research were opened up. Examples of the mostinteresting of these formulae include the intriguing infinite series for, one of which is givenbelow


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\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 = 458 with classnumber h(d) = 2 (note that 571358 = 26390 and that 9801=9999; 396=499) and is related tothe 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 converges extraordinarily rapidly (exponentially) and forms thebasis of some of the fastest algorithms currently used to calculate . Truncating thesum to the first term also gives the approximation 9801\sqrt{2}/4412 for , which iscorrect to six decimal places.

One of his remarkable capabilities was the rapid solution for problems. He was sharinga room with P. C. Mahalanobis who had a problem, "Imagine that you are on a streetwith houses marked 1 through n. There is a house in between (x) such that the sum ofthe house numbers to left of it equals the sum of the house numbers to its right. If n isbetween 50 and 500, what are n and x?" This is a bivariate problem with multiplesolutions. Ramanujan thought about it and gave the answer with a twist: He gave acontinued fraction. The unusual part was that it was the solution to the whole class ofproblems. Mahalanobis was astounded and asked how he did it. "It is simple. Theminute I heard the problem, I knew that the answer was a continued fraction. Whichcontinued fraction, I asked myself. Then the answer came to my mind," Ramanujanreplied.[85][86]

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 \theta, where \Gamma(z) is the gamma function. Expanding into series of powersand equating coefficients of \theta^0, \theta^4, and \theta^8 gives some deepidentities for the hyperbolic secant.

In 1918, Hardy and Ramanujan studied the partition function P(n) extensively and gavea non-convergent asymptotic series that permits exact computation of the number ofpartitions of an integer. Hans Rademacher, in 1937, was able to refine their formula tofind an exact convergent series solution to this problem. Ramanujan and Hardy's workin this area gave rise to a powerful new method for finding asymptotic formulae, calledthe circle method.[87]

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

harmonic weak Maass forms.The Ramanujan conjectureMain article: RamanujanPetersson conjecture

Although there are numerous statements that could bear the name Ramanujanconjecture, there is one statement that was very influential on later work. In particular,the connection of this conjecture with conjectures of Andr Weil in algebraic geometryopened up new areas of research. That Ramanujan conjecture is an assertion on the size of the taufunction, which has as generating function the discriminant modular form (q), a typical cusp


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form in the theory of modular forms. It was finally proven in 1973, as a consequence ofPierre Deligne's proof of the Weil conjectures. The reduction step involved iscomplicated. Deligne won a Fields Medal in 1978 for his work on Weil conjectures.[88]Ramanujan's notebooksFurther information: Ramanujan's lost notebook

While still in Madras, Ramanujan recorded the bulk of his results in four notebooks ofloose leaf paper. These results were mostly written up without any derivations. This isprobably the origin of the misperception that Ramanujan was unable to prove hisresults 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 mostcertainly 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 veryexpensive, Ramanujan would do most of his work and perhaps his proofs on slate, andthen transfer just the results to paper. Using a slate was common for mathematicsstudents in the Madras Presidency at the time. He was also quite likely to have beeninfluenced by the style of G. S. Carr's book studied in his teenage, which stated resultswithout proofs. Finally, it is possible that Ramanujan considered his workings to be forhis personal interest alone; and therefore recorded only the results.[89]

The first notebook has 351 pages with 16 somewhat organized chapters and someunorganized material. The second notebook has 256 pages in 21 chapters and 100unorganised pages, with the third notebook containing 33 unorganised pages. Theresults in his notebooks inspired numerous papers by later mathematicians trying toprove what he had found. Hardy himself created papers exploring material fromRamanujan's work as did G. N. Watson, B. M. Wilson, and Bruce Berndt.[89] A fourthnotebook with 87 unorganised pages, the so-called "lost notebook", was rediscoveredin 1976 by George Andrews.[3]RamanujanHardy number 1729Main article: 1729 (number)

A common anecdote about Ramanujan relates to the number 1729. Hardy arrived atRamanujan's residence in a cab numbered 1729. Hardy commented that the number1729 seemed to be uninteresting. Ramanujan is said to have stated on the spot that itwas actually a very interesting number mathematically, being the smallest naturalnumber representable in two different ways as a sum of two positive cubes:

1729=1^3+12^3=9^3+10^3.\,

Generalizations of this idea have created the notion of "taxicab numbers".Coincidentally, 1729 is also a Carmichael Number.Other mathematicians' views of Ramanujan

Hardy said : "The limitations of his knowledge were as startling as its profundity. Herewas 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 inthe world, who had found for himself the functional equation of the zeta function andthe dominant terms of many of the most famous problems in the analytic theory ofnumbers; and yet he had never heard of a doubly periodic function or of Cauchy'stheorem, and had indeed but the vaguest idea of what a function of a complex variablewas...".[90] When asked about the methods employed by Ramanujan to arrive at hissolutions, Hardy said that they were "arrived at by a process of mingled argument,intuition, and induction, of which he was entirely unable to give any coherentaccount."[91] He also stated that he had "never met his equal, and can compare him


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only with Euler or Jacobi."[91]

Quoting K. Srinivasa Rao,[92] "As for his place in the world of Mathematics, we quoteBruce C. Berndt: 'Paul Erd s has passed on to us Hardy's personal ratings of mathematicians. Suppose that we rate mathematicians on the basis of pure talent on ascale 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 spoke of "SrinivasaRamanujan, discovered by the Cambridge mathematician Hardy, whose greatmathematical findings were beginning to be appreciated from 1915 to 1919. Hisachievements were to be fully understood much later, well after his untimely death in1920. For example, his work on the highly composite numbers (numbers with a largenumber of factors) started a whole new line of investigations in the theory of suchnumbers."

During his lifelong mission in educating and propagating mathematics among theschool children in India, Nigeria and elsewhere, P.K. Srinivasan has continuallyintroduced Ramanujan's mathematical works.Recognition

Ramanujan's home state of Tamil Nadu celebrates 22 December (Ramanujan'sbirthday) as 'State IT Day', memorializing both the man and his achievements, as anative of Tamil Nadu. A stamp picturing Ramanujan was released by the Governmentof India in 1962 the 75th anniversary of Ramanujan's birth commemorating hisachievements in the field of number theory.

Since the Centennial year of Srinivasa Ramanujan, every year 22 Dec, is celebrated asRamanujan Day by the Government Arts College, Kumbakonam where he had studiedand later dropped out. It is celebrated by the Department Of Mathematics byorganising one-, two-, or three-day seminar by inviting eminent scholars fromuniversities/colleges, and participants are mainly students of Mathematics, researchscholars, and professors from local colleges. It has been planned to celebrate the 125-th birthday in a grand manner by inviting the foreign Eminent Mathematical scholars ofthis century viz., G E Andrews. and Bruce C Berndt, who are very familiar with thecontributions and works of Ramanujan.

Every year, in Chennai (formerly Madras), the Indian Institute of Technology (IIT),Ramanujan's work and life are celebrated on 22 December. The Department ofMathematics celebrates this day by organising a National Symposium On MathematicalMethods and Applications (NSMMA) for one day by inviting Eminent scholars from Indiaand foreign countries.

A prize for young mathematicians from developing countries has been created in thename of Ramanujan by the International Centre for Theoretical Physics (ICTP), incooperation with the International Mathematical Union, who nominate members of the

prize committee. The Shanmugha Arts, Science, Technology & Research Academy(SASTRA), based in the state of Tamil Nadu in South India, has instituted the SASTRARamanujan Prize of $10,000 to be given annually to a mathematician not exceedingthe age of 32 for outstanding contributions in an area of mathematics influenced byRamanujan. The age limit refers to the years Ramanujan lived, having nevertheless stillachieved many accomplishments. This prize has been awarded annually since 2005, atan international conference conducted by SASTRA in Kumbakonam, Ramanujan'shometown, around Ramanujan's birthday, 22 December.


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On the 125th anniversary of his birth, India declared the birthday of Ramanujan,December 22, as 'National Mathematics Day.' The declaration was made by Dr.Manmohan Singh in Chennai on December 26, 2011.[93] Dr Manmohan Singh alsodeclared that the year 2012 would be celebrated as the National Mathematics Year.

Documents

Ramanujan srinivasa