Srinivasa Ramanujan

“Ramanujan” redirects here. For other uses, seeRamanujan (disambiguation).In this Indian name, the name Srinivasa is a patronymic,not a family name, and the person should be referred toby the given name, Ramanujan.

Srinivasa Ramanujan FRS (pronunciation:i/sriː.ni.vɑː.sərɑː.mɑː.nʊ.dʒən/) (22 December 1887

– 26 April 1920) was an Indian mathematician andautodidact who, with almost no formal training inpure mathematics, made extraordinary contributions tomathematical analysis, number theory, infinite series,and continued fractions. Ramanujan initially developedhis own mathematical research in isolation, which wasquickly recognized by Indian mathematicians. Whenhis skills became apparent to the wider mathematicalcommunity, centered in Europe at the time, he began afamous partnership with the English mathematician G.H. Hardy. He rediscovered previously known theoremsin addition to producing new work. Ramanujan wassaid to be a natural genius, in the same league asmathematicians such as Euler and Gauss.[1]

During his short life, Ramanujan independently compilednearly 3900 results (mostly identities and equations).[2]Nearly all his claims have now been proven correct, al-though a small number of these results were actually falseand some were already known.[3] He stated results thatwere both original and highly unconventional, such as theRamanujan prime and the Ramanujan theta function, andthese have inspired a vast amount of further research.[4]The Ramanujan Journal, an international publication,was launched to publish work in all areas of mathematicsinfluenced by his work.[5]

1 Early life

Ramanujan was born on 22 December 1887 in Erode,Madras Presidency (now Pallipalayam, Erode, TamilNadu), at the residence of his maternal grandparents.[6]His father, K. Srinivasa Iyengar, worked as a clerk in asari shop and hailed from the district of Thanjavur.[7] Hismother, Komalatammal, was a housewife and also sangat a local temple.[8] They lived in Sarangapani Street ina traditional home in the town of Kumbakonam. Thefamily home is now a museum. When Ramanujan was ayear and a half old, his mother gave birth to a son namedSadagopan, who died less than three months later. In De-cember 1889, Ramanujan had smallpox and recovered,

Ramanujan’s home on Sarangapani Street, Kumbakonam

unlike thousands in the Thanjavur District who died fromthe disease that year.[9] He moved with his mother toher parents’ house in Kanchipuram, near Madras (nowChennai). In November 1891, and again in 1894, hismother gave birth to two children, but both children diedin infancy.On 1 October 1892, Ramanujan was enrolled at the lo-cal school.[10] In March 1894, he was moved to a Telugumedium school. After his maternal grandfather lost hisjob as a court official in Kanchipuram,[11] Ramanujanand his mother moved back to Kumbakonam and he wasenrolled in the Kangayan Primary School.[12] When hispaternal grandfather died, he was sent back to his mater-nal grandparents, who were now living in Madras. He didnot like school in Madras, and he tried to avoid attending.His family enlisted a local constable to make sure he at-tended school. Within six months, Ramanujan was backin Kumbakonam.[12]

Since Ramanujan’s father was at work most of the day,his mother took care of him as a child. He had a closerelationship with her. From her, he learned about tra-dition and puranas. He learned to sing religious songs,

1

2 2 ADULTHOOD IN INDIA

to attend pujas at the temple and particular eating habits– all of which are part of Brahmin culture.[13] At theKangayan Primary School, Ramanujan performed well.Just before the age of 10, in November 1897, he passedhis primary examinations in English, Tamil, geographyand arithmetic. With his scores, he stood first in thedistrict.[14] That year, Ramanujan entered Town HigherSecondary School where he encountered formal mathe-matics for the first time.[14]

By age 11, he had exhausted the mathematical knowledgeof two college students who were lodgers at his home. Hewas later lent a book on advanced trigonometry written byS. L. Loney.[15][16] He completely mastered this book bythe age of 13 and discovered sophisticated theorems onhis own. By 14, he was receiving merit certificates andacademic awards which continued throughout his schoolcareer and also assisted the school in the logistics of as-signing its 1200 students (each with their own needs) toits 35-odd teachers.[17] He completed mathematical ex-ams in half the allotted time, and showed a familiaritywith geometry and infinite series. Ramanujan was shownhow to solve cubic equations in 1902 and he went on tofind his own method to solve the quartic. The followingyear, not knowing that the quintic could not be solved byradicals, he tried (and of course failed) to solve the quin-tic.In 1903 when he was 16, Ramanujan obtained froma friend a library-loaned copy of a book by G. S.Carr.[18][19] The book was titledA Synopsis of ElementaryResults in Pure and Applied Mathematics and was a col-lection of 5000 theorems. Ramanujan reportedly studiedthe contents of the book in detail.[20] The book is gen-erally acknowledged as a key element in awakening thegenius of Ramanujan.[20] The next year, he had indepen-dently developed and investigated the Bernoulli numbersand had calculated the Euler–Mascheroni constant up to15 decimal places.[21] His peers at the time commentedthat they “rarely understood him” and “stood in respectfulawe” of him.[17]

When he graduated from Town Higher Secondary Schoolin 1904, Ramanujan was awarded the K. RanganathaRao prize for mathematics by the school’s headmas-ter, Krishnaswami Iyer. Iyer introduced Ramanujanas an outstanding student who deserved scores higherthan the maximum possible marks.[17] He received ascholarship to study at Government Arts College, Kum-bakonam,[22][23] However, Ramanujan was so intent onstudying mathematics that he could not focus on anyother subjects and failed most of them, losing his schol-arship in the process.[24] In August 1905, he ran awayfrom home, heading towards Visakhapatnam and stayedin Rajahmundry[25] for about a month.[26] He later en-rolled at Pachaiyappa’s College in Madras. He again ex-celled in mathematics but performed poorly in other sub-jects such as physiology. Ramanujan failed his Fellowof Arts exam in December 1906 and again a year later.Without a degree, he left college and continued to pur-

sue independent research in mathematics. At this pointin his life, he lived in extreme poverty and was often onthe brink of starvation.[27]

2 Adulthood in India

On 14 July 1909, Ramanujan was married to a ten-year old bride, Janakiammal (21 March 1899 – 13 April1994).[28] She came from Rajendram, a village close toMarudur (Karur district) Railway Station. Ramanujan’sfather did not participate in the marriage ceremony.[29]

After the marriage, Ramanujan developed a hydroceletestis, an abnormal swelling of the tunica vaginalis, an in-ternal membrane in the testicle.[30] The condition couldbe treated with a routine surgical operation that wouldrelease the blocked fluid in the scrotal sac. His familydid not have the money for the operation, but in January1910, a doctor volunteered to do the surgery for free.[31]

After his successful surgery, Ramanujan searched for ajob. He stayed at friends’ houses while he went door todoor around the city of Madras (now Chennai) lookingfor a clerical position. To make some money, he tutoredsome students at Presidency College who were preparingfor their F.A. exam.[32]

In late 1910, Ramanujan was sick again, possibly as a re-sult of the surgery earlier in the year. He feared for hishealth, and even told his friend, R. Radakrishna Iyer, to“hand these [Ramanujan’s mathematical notebooks] overto Professor Singaravelu Mudaliar [the mathematics pro-fessor at Pachaiyappa’s College] or to the British profes-sor Edward B. Ross, of theMadras Christian College.”[33]After Ramanujan recovered and got back his notebooksfrom Iyer, he took a northbound train fromKumbakonamto Villupuram, a coastal city under French control.[34][35]

2.1 Attention towards mathematics

Ramanujan met deputy collector V. Ramaswamy Aiyer,who had recently founded the Indian MathematicalSociety.[36] Ramanujan, wishing for a job at the revenuedepartment where Ramaswamy Aiyer worked, showedhim his mathematics notebooks. As Ramaswamy Aiyerlater recalled:

I was struck by the extraordinary mathe-matical results contained in it [the notebooks].I had no mind to smother his genius by an ap-pointment in the lowest rungs of the revenuedepartment.[37]

Ramaswamy Aiyer sent Ramanujan, with letters of in-troduction, to his mathematician friends in Madras.[36]Some of these friends looked at his work and gave himletters of introduction to R. Ramachandra Rao, the dis-trict collector for Nellore and the secretary of the Indian

2.2 Contacting English mathematicians 3

Mathematical Society.[38][39][40] Ramachandra Rao wasimpressed by Ramanujan’s research but doubted that itwas actually his own work. Ramanujan mentioned a cor-respondence he had with Professor Saldhana, a notableBombay mathematician, in which Saldhana expressed alack of understanding of his work but concluded thathe was not a phoney.[41] Ramanujan’s friend, C. V. Ra-jagopalachari, persisted with Ramachandra Rao and triedto quell any doubts over Ramanujan’s academic integrity.Rao agreed to give him another chance, and he listened asRamanujan discussed elliptic integrals, hypergeometricseries, and his theory of divergent series, which Rao saidultimately “converted” him to a belief in Ramanujan’smathematical brilliance.[41] When Rao asked him whathe wanted, Ramanujan replied that he needed some workand financial support. Rao consented and sent him toMadras. He continued his mathematical research withRao’s financial aid taking care of his daily needs. Ra-manujan, with the help of Ramaswamy Aiyer, had hiswork published in the Journal of the Indian Mathemati-cal Society.[42]

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

√1 + 2

√1 + 3

√1 + · · ·.

He waited for a solution to be offered in three issues, oversixmonths, but failed to receive any. At the end, Ramanu-jan supplied the solution to the problem himself. On page105 of his first notebook, he formulated an equation thatcould be used to solve the infinitely nested radicals prob-lem.

x+n+a =

√ax+ (n+ a)2 + x

√a(x+ n) + (n+ a)2 + (x+ n)

√· · ·

Using this equation, the answer to the question posedin the Journal was simply 3.[43] Ramanujan wrote hisfirst formal paper for the Journal on the properties ofBernoulli numbers. One property he discovered was thatthe denominators (sequence A027642 in OEIS) of thefractions of Bernoulli numbers were always divisible bysix. He also devised a method of calculating Bn based onprevious Bernoulli numbers. One of these methods wentas follows:It will be observed that if n is even but not equal to zero,(i) Bn is a fraction and the numerator of Bn

n in its lowestterms is a prime number,(ii) the denominator of Bn contains each of the factors 2and 3 once and only once,(iii) 2n(2n − 1) bnn is an integer and 2(2n − 1)Bn con-sequently is an odd integer.In his 17-page paper, “Some Properties of Bernoulli’sNumbers”, Ramanujan gave three proofs, two corollariesand three conjectures.[44] Ramanujan’s writing initiallyhad many flaws. As Journal editor M. T. Narayana Iyen-gar noted:

Mr. Ramanujan’s methods were so terseand novel and his presentation so lackingin clearness and precision, that the ordinary[mathematical reader], unaccustomed to suchintellectual gymnastics, could hardly followhim.[45]

Ramanujan later wrote another paper and also continuedto provide problems in the Journal.[46] In early 1912, hegot a temporary job in the Madras Accountant General'soffice, with a salary of 20 rupees per month. He lasted foronly a few weeks.[47] Toward the end of that assignmenthe applied for a position under the Chief Accountant ofthe Madras Port Trust. In a letter dated 9 February 1912,Ramanujan wrote:

Sir,I understand there is a clerkship vacant inyour office, and I beg to apply for the same.I have passed the Matriculation Examinationand studied up to the F.A. but was preventedfrom pursuing my studies further owing to sev-eral untoward circumstances. I have, however,been devoting all my time to Mathematics anddeveloping the subject. I can say I am quiteconfident I can do justice to my work if I amappointed to the post. I therefore beg to re-quest that you will be good enough to conferthe appointment on me.[48]

Attached to his application was a recommendationfrom E. W. Middlemast, a mathematics professor atthe Presidency College, who wrote that Ramanujanwas “a young man of quite exceptional capacity inMathematics”.[49] Three weeks after he had applied, on1 March, Ramanujan learned that he had been acceptedas a Class III, Grade IV accounting clerk, making 30 ru-pees per month.[50] At his office, Ramanujan easily andquickly completed the work he was given, so he spent hisspare time doing mathematical research. Ramanujan’sboss, Sir Francis Spring, and S. Narayana Iyer, a col-league who was also treasurer of the Indian Mathemat-ical Society, encouraged Ramanujan in his mathematicalpursuits.

2.2 Contacting English mathematicians

In the spring of 1913, Narayana Iyer, Ramachandra Raoand E.W.Middlemast tried to present Ramanujan’s workto British mathematicians. One mathematician, M. J. M.Hill of University College London, commented that Ra-manujan’s papers were riddled with holes.[51] He said thatalthough Ramanujan had “a taste for mathematics, andsome ability”, he lacked the educational background andfoundation needed to be accepted by mathematicians.[52]Although Hill did not offer to take Ramanujan on as astudent, he did give thorough and serious professional ad-vice on his work. With the help of friends, Ramanujan

4 3 LIFE IN ENGLAND

drafted letters to leading mathematicians at CambridgeUniversity.[53]

The first two professors, H. F. Baker and E. W. Hob-son, returned Ramanujan’s papers without comment.[54]On 16 January 1913, Ramanujan wrote to G. H. Hardy.Coming from an unknown mathematician, the nine pagesof mathematics made Hardy initially view Ramanu-jan’s manuscripts as a possible “fraud”.[55] Hardy recog-nised some of Ramanujan’s formulae but others “seemedscarcely possible to believe”.[56] One of the theoremsHardy found scarcely possible to believe was found onthe bottom of page three (valid for 0 < a < b + 1/2):

∫ ∞

0

1 + x2/(b+ 1)2

1 + x2/(a)2×1 + x2/(b+ 2)2

1 + x2/(a+ 1)2×· · · dx =

√π

2×Γ(a+ 1

2 )Γ(b+ 1)Γ(b− a+ 12 )

Γ(a)Γ(b+ 12 )Γ(b− a+ 1)

.

Hardy was also impressed by some of Ramanujan’s otherwork relating to infinite series:

1−5

(1

2

)3

+9

(1× 3

2× 4

)3

−13

(1× 3× 5

2× 4× 6

)3

+· · · = 2

π

1+9

(1

4

)4

+17

(1× 5

4× 8

)4

+25

(1× 5× 9

4× 8× 12

)4

+· · · = 232

π12Γ2

(34

) .The first result had already been determined by a math-ematician named Bauer. The second one was new toHardy, and was derived from a class of functions calleda hypergeometric series which had first been researchedby Leonhard Euler and Carl Friedrich Gauss. Comparedto Ramanujan’s work on integrals, Hardy found these re-sults “much more intriguing”.[57] After he saw Ramanu-jan’s theorems on continued fractions on the last pageof the manuscripts, Hardy commented that “they [the-orems] defeated me completely; I had never seen any-thing in the least like them before”.[58] He figured thatRamanujan’s theorems “must be true, because, if theywere not true, no one would have the imagination to in-vent them”.[58] Hardy asked a colleague, J. E. Littlewood,to take a look at the papers. Littlewood was amazed bythe mathematical genius of Ramanujan. After discussingthe papers with Littlewood, Hardy concluded that the let-ters were “certainly the most remarkable I have received”and commented that Ramanujan was “a mathematicianof the highest quality, a man of altogether exceptionaloriginality and power”.[59] One colleague, E. H. Neville,later commented that “not one [theorem] could have beenset in the most advanced mathematical examination in theworld”.[60]

On 8 February 1913, Hardy wrote a letter to Ramanujan,expressing his interest for his work. Hardy also addedthat it was “essential that I should see proofs of some ofyour assertions”.[61] Before his letter arrived in Madrasduring the third week of February, Hardy contacted theIndian Office to plan for Ramanujan’s trip to Cambridge.Secretary Arthur Davies of the Advisory Committee for

Indian Students met with Ramanujan to discuss the over-seas trip.[62] In accordance with his Brahmin upbring-ing, Ramanujan refused to leave his country to “go toa foreign land”.[63] Meanwhile, Ramanujan sent a letterpacked with theorems to Hardy, writing, “I have found afriend in you who views my labour sympathetically.”[64]

To supplement Hardy’s endorsement, a former mathe-matical lecturer at Trinity College, Cambridge, GilbertWalker, looked at Ramanujan’s work and expressedamazement, urging him to spend time at Cambridge.[65]As a result of Walker’s endorsement, B. HanumanthaRao, a mathematics professor at an engineering college,invited Ramanujan’s colleague Narayana Iyer to a meet-ing of the Board of Studies in Mathematics to discuss“what we can do for S. Ramanujan”.[66] The board agreedto grant Ramanujan a research scholarship of 75 ru-pees per month for the next two years at the Universityof Madras.[67] While he was engaged as a research stu-dent, Ramanujan continued to submit papers to the Jour-nal of the Indian Mathematical Society. In one instance,Narayana Iyer submitted some theorems of Ramanujanon summation of series to the abovemathematical journaladding “The following theorem is due to S. Ramanujan,the mathematics student of Madras University”. Later inNovember, British Professor Edward B. Ross of MadrasChristian College, whom Ramanujan had met a few yearsbefore, stormed into his class one day with his eyes glow-ing, asking his students, “Does Ramanujan know Polish?"The reason was that in one paper, Ramanujan had antic-ipated the work of a Polish mathematician whose paperhad just arrived by the day’s mail.[68] In his quarterly pa-pers, Ramanujan drew up theorems to make definite in-tegrals more easily solvable. Working off Giuliano Frul-lani’s 1821 integral theorem, Ramanujan formulated gen-eralisations that could be made to evaluate formerly un-yielding integrals.[69]

Hardy’s correspondence with Ramanujan soured afterRamanujan refused to come to England. Hardy enlisteda colleague lecturing in Madras, E. H. Neville, to mentorand bring Ramanujan to England.[70] Neville asked Ra-manujan why he would not go to Cambridge. Ramanu-jan apparently had now accepted the proposal; as Nevilleput it, “Ramanujan needed no converting and that hisparents’ opposition had been withdrawn”.[60] Apparently,Ramanujan’s mother had a vivid dream in which the fam-ily Goddess, the deity of Namagiri, commanded her “tostand no longer between her son and the fulfilment of hislife’s purpose”.[60] Ramanujan then set sail for England,leaving his wife to stay with his parents in India.

3 Life in England

Ramanujan boarded the S.S. Nevasa on 17 March 1914,and at 10 o'clock in the morning, the ship departed fromMadras.[71] He arrived in London on 14 April, with E.H. Neville waiting for him with a car. Four days later,

3.1 Illness and return to India 5

Ramanujan (centre) with other scientists at Trinity College

Whewell’s Court, Trinity College, Cambridge

Neville took him to his house on Chesterton Road inCambridge. Ramanujan immediately began his workwith Littlewood and Hardy. After six weeks, Ramanu-jan moved out of Neville’s house and took up residenceonWhewell’s Court, just a five-minute walk fromHardy’sroom.[72] Hardy and Ramanujan began to take a look atRamanujan’s notebooks. Hardy had already received 120theorems from Ramanujan in the first two letters, butthere were many more results and theorems to be foundin the notebooks. Hardy saw that some were wrong,others had already been discovered, while the rest werenew breakthroughs.[73] Ramanujan left a deep impres-sion on Hardy and Littlewood. Littlewood commented,“I can believe that he’s at least a Jacobi",[74] while Hardysaid he “can compare him only with [Leonhard] Euler orJacobi.”[75]

Ramanujan spent nearly five years in Cambridge collab-orating with Hardy and Littlewood and published a partof his findings there. Hardy and Ramanujan had highlycontrasting personalities. Their collaboration was a clashof different cultures, beliefs and working styles. Hardywas an atheist and an apostle of proof and mathematicalrigour, whereas Ramanujan was a deeply religious manand relied very strongly on his intuition. While in Eng-land, Hardy tried his best to fill the gaps in Ramanujan’seducation without interrupting his spell of inspiration.Ramanujan was awarded a Bachelor of Science degree byresearch (this degree was later renamed PhD) in March1916 for his work on highly composite numbers, the firstpart of which was published as a paper in the Proceed-ings of the London Mathematical Society. The paper wasover 50 pages with different properties of such numbersproven. Hardy remarked that this was one of the most un-usual papers seen in mathematical research at that timeand that Ramanujan showed extraordinary ingenuity inhandling it. On 6 December 1917, he was elected to theLondon Mathematical Society. He became a Fellow ofthe Royal Society in 1918, becoming the second Indian todo so, following Ardaseer Cursetjee in 1841, and he wasone of the youngest Fellows in the history of the RoyalSociety. He was elected “for his investigation in Ellipticfunctions and the Theory of Numbers.” On 13 October1918, he became the first Indian to be elected a Fellow ofTrinity College, Cambridge.[76]

3.1 Illness and return to India

Plagued by health problems throughout his life, livingin a country far away from home, and obsessively in-volved with his mathematics, Ramanujan’s health wors-ened in England, perhaps exacerbated by stress and by thescarcity of vegetarian food during the First World War.He was diagnosed with tuberculosis and a severe vitamindeficiency and was confined to a sanatorium.Ramanujan returned to Kumbakonam, Madras Presi-dency in 1919 and died soon thereafter at the age of 32in 1920. His widow, S. Janaki Ammal, moved to Mum-bai, but returned to Chennai (formerly Madras) in 1950,where she lived until her death at age 94 in 1994.[29]

A 1994 analysis of Ramanujan’s medical records andsymptoms by Dr. D.A.B. Young concluded that it wasmuch more likely he had hepatic amoebiasis, a para-sitic infection of the liver widespread in Madras, whereRamanujan had spent time. He had two episodes ofdysentery before he left India. When not properly treated,dysentery can lie dormant for years and lead to hepaticamoebiasis,[77] a difficult disease to diagnose, but oncediagnosed readily cured.[77]

6 4 MATHEMATICAL ACHIEVEMENTS

3.2 Personality and spiritual life

Ramanujan has been described as a person with a some-what shy and quiet disposition, a dignifiedmanwith pleas-ant manners.[78] He lived a rather Spartan life while atCambridge. Ramanujan’s first Indian biographers de-scribe him as rigorously orthodox. Ramanujan cred-ited his acumen to his family goddess, Mahalakshmiof Namakkal. He looked to her for inspiration in hiswork,[79] and claimed to dream of blood drops that sym-bolised her male consort, Narasimha, after which hewould receive visions of scrolls of complex mathemati-cal content unfolding before his eyes.[80] He often said,“An equation for me has no meaning, unless it representsa thought of God.”[81][82]

Hardy cites Ramanujan as remarking that all religionsseemed equally true to him.[83] Hardy further arguedthat Ramanujan’s religiousness had been romanticised byWesterners and overstated—in reference to his belief, notpractice—by Indian biographers. At the same time, heremarked on Ramanujan’s strict observance of vegetari-anism.

4 Mathematical achievements

In mathematics, there is a distinction between having aninsight and having a proof. Ramanujan’s talent suggesteda plethora of formulae that could then be investigated indepth later. It is said by G. H. Hardy that Ramanujan’sdiscoveries are unusually rich and that there is often moreto them than initially meets the eye. As a by-product, newdirections of research were opened up. Examples of themost interesting of these formulae include the intriguinginfinite series for π, one of which is given below

1

π=

2√2

9801

∞∑k=0

(4k)!(1103 + 26390k)

(k!)43964k.

This result is based on the negative fundamental discrim-inant d = −4×58 = −232 with class number h(d) = 2(note that 5×7×13×58 = 26390 and that 9801=99×99;396=4×99) and is related to the fact that

eπ√58 = 3964 − 104.000000177 . . . .

Compare to Heegner numbers, which have class num-ber 1 and yield similar formulae. Ramanujan’s series forπ converges extraordinarily rapidly (exponentially) andforms the basis of some of the fastest algorithms currentlyused to calculate π. Truncating the sum to the first termalso gives the approximation 9801

√2/4412 for π, which

is correct to six decimal places. See also the more generalRamanujan–Sato series.

One of his remarkable capabilities was the rapid solutionfor problems. He was sharing a room with P. C. Maha-lanobis who had a problem, “Imagine that you are on astreet with houses marked 1 through n. There is a housein between (x) such that the sum of the house numbers toleft 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 isa bivariate problem with multiple solutions. Ramanujanthought about it and gave the answer with a twist: He gavea continued fraction. The unusual part was that it was thesolution to the whole class of problems. Mahalanobis wasastounded and asked how he did it. “It is simple. Theminute I heard the problem, I knew that the answer wasa continued fraction. Which continued fraction, I askedmyself. Then the answer came to my mind”, Ramanujanreplied.[84][85]

His intuition also led him to derive some previously un-known identities, such as

[1 + 2

∞∑n=1

cos(nθ)cosh(nπ)

]−2

+

[1 + 2

∞∑n=1

cosh(nθ)cosh(nπ)

]−2

=2Γ4

(34

)π

for all θ , where Γ(z) is the gamma function. Expand-ing into series of powers and equating coefficients of θ0 ,θ4 , and θ8 gives some deep identities for the hyperbolicsecant.In 1918, Hardy and Ramanujan studied the partitionfunction P(n) extensively and gave a non-convergentasymptotic series that permits exact computation of thenumber of partitions of an integer. Hans Rademacher,in 1937, was able to refine their formula to find an exactconvergent series solution to this problem. Ramanujanand Hardy’s work in this area gave rise to a powerful newmethod for finding asymptotic formulae, called the circlemethod.[86]

He discovered mock theta functions in the last year of hislife.[87] For many years these functions were a mystery,but they are now known to be the holomorphic parts ofharmonic weak Maass forms.

4.1 The Ramanujan conjecture

Main article: Ramanujan–Petersson conjecture

Although there are numerous statements that could haveborne the name Ramanujan conjecture, there is one state-ment that was very influential on later work. In particu-lar, the connection of this conjecture with conjectures ofAndréWeil in algebraic geometry opened up new areas ofresearch. That Ramanujan conjecture is an assertion onthe size of the tau-function, which has as generating func-tion the discriminant modular form Δ(q), a typical cuspform in the theory of modular forms. It was finally provenin 1973, as a consequence of Pierre Deligne's proof of the

7

Weil conjectures. The reduction step involved is compli-cated. Deligne won a Fields Medal in 1978 for his workon Weil conjectures.[88]

4.2 Ramanujan’s notebooks

Further information: Ramanujan’s lost notebook

While still in Madras, Ramanujan recorded the bulk ofhis results in four notebooks of loose leaf paper. Theseresults were mostly written up without any derivations.This is probably the origin of the misperception thatRamanujan was unable to prove his results and simplythought up the final result directly. Mathematician BruceC. Berndt, in his review of these notebooks and Ramanu-jan’s work, says that Ramanujan most certainly was ableto make the proofs of most of his results, but chose notto.This style of working may have been for several reasons.Since paper was very expensive, Ramanujan would domost of his work and perhaps his proofs on slate, andthen transfer just the results to paper. Using a slate wascommon for mathematics students in the Madras Presi-dency at the time. He was also quite likely to have beeninfluenced by the style of G. S. Carr's book studied inhis youth, which stated results without proofs. Finally,it is possible that Ramanujan considered his workings tobe for his personal interest alone; and therefore recordedonly the results.[89]

The first notebook has 351 pages with 16 somewhat or-ganised chapters and some unorganised material. Thesecond notebook has 256 pages in 21 chapters and 100unorganised pages, with the third notebook containing 33unorganised pages. The results in his notebooks inspirednumerous papers by later mathematicians trying to provewhat he had found. Hardy himself created papers explor-ingmaterial fromRamanujan’s work as did G. N.Watson,B. M. Wilson, and Bruce Berndt.[89] A fourth notebookwith 87 unorganised pages, the so-called “lost notebook”,was rediscovered in 1976 by George Andrews.[77]

Notebooks 1, 2 and 3 were published as a two-volume setin 1957 by the Tata Institute of Fundamental Research(TIFR), Mumbai, India. This was a photocopy edition ofthe original manuscripts, in his own handwriting.In December 2011, as part of the celebrations of the125th anniversary of Ramanujan’s birth, TIFR repub-lished the notebooks in a coloured two-volume collector’sedition. These were produced from scanned and mi-crofilmed images of the original manuscripts by expertarchivists of Roja Muthiah Research Library, Chennai.

5 Hardy-Ramanujan number 1729

Main article: 1729 (number)

The number 1729 is known as the Hardy–Ramanujannumber after a famous anecdote of the British mathe-matician G. H. Hardy regarding a visit to the hospital tosee Ramanujan. In Hardy’s words:[90]

The two different ways are

1729 = 13 + 123 = 93 + 103.

Generalizations of this idea have created the notionof "taxicab numbers". Coincidentally, 1729 is also aCarmichael number.

6 Other mathematicians’ views ofRamanujan

Hardy said : “He combined a power of generalization, afeeling for form, and a capacity for rapid modification ofhis hypotheses, that were often really startling, and madehim, in his own peculiar field, without a rival in his day.The limitations of his knowledge were as startling as itsprofundity. Here was a man who could work out modularequations and theorems... to orders unheard of, whosemastery of continued fractions was... beyond that of anymathematician in the world, who had found for himselfthe functional equation of the zeta function and the dom-inant terms of many of the most famous problems in theanalytic theory of numbers; and yet he had never heardof a doubly periodic function or of Cauchy’s theorem,and had indeed but the vaguest idea of what a functionof a complex variable was...”.[91] When asked about themethods employed by Ramanujan to arrive at his solu-tions, Hardy said that they were “arrived at by a processof mingled argument, intuition, and induction, of whichhe was entirely unable to give any coherent account.”[92]He also stated that he had “never met his equal, and cancompare him only with Euler or Jacobi.”[92]

Quoting K. Srinivasa Rao,[93] “As for his place in theworld of Mathematics, we quote Bruce C. Berndt: 'PaulErdős has passed on to us Hardy’s personal ratings ofmathematicians. Suppose that we rate mathematicians onthe basis of pure talent on a scale from 0 to 100, Hardygave himself a score of 25, J.E. Littlewood 30, DavidHilbert 80 and Ramanujan 100.'"Professor Bruce C. Berndt of the University of Illinois,during a lecture at IIT Madras in May 2011, stated thatover the last 40 years, as nearly all of Ramanujan’s theo-rems have been proven right, there had been a greater ap-preciation of Ramanujan’s work and brilliance. Further,he stated Ramanujan’s work was now pervading many ar-eas of modern mathematics and physics.[87][94]

8 8 IN POPULAR CULTURE

In his book Scientific Edge, the physicist Jayant Narlikarspoke of “Srinivasa Ramanujan, discovered by the Cam-bridge mathematician Hardy, whose great mathematicalfindings were beginning to be appreciated from 1915 to1919. His achievements were to be fully understoodmuch later, well after his untimely death in 1920. For ex-ample, his work on the highly composite numbers (num-bers with a large number of factors) started a whole newline of investigations in the theory of such numbers.”During his lifelong mission in educating and propagatingmathematics among the school children in India, Nige-ria and elsewhere, P.K. Srinivasan has continually intro-duced Ramanujan’s mathematical works.

7 Recognition

Further information: List of things named after SrinivasaRamanujanRamanujan’s home state of Tamil Nadu celebrates 22

Bust of Ramanujan in the garden of Birla Industrial & Techno-logical Museum.

December (Ramanujan’s birthday) as 'State IT Day',memorialising both the man and his achievements, as anative of Tamil Nadu. A stamp picturing Ramanujan wasreleased by the Government of India in 1962 – the 75thanniversary of Ramanujan’s birth – commemorating hisachievements in the field of number theory,[95] and a newdesign was issued on 26 December 2011, by the IndiaPost.[96][97]

Since the Centennial year of Ramanujan, every year 22

Dec, is celebrated as Ramanujan Day by the GovernmentArts College, Kumbakonam where he had studied andlater dropped out. It is celebrated by the Departmentof Mathematics by organising one-, two-, or three-dayseminars by inviting eminent scholars from universi-ties/colleges, and participants are mainly students ofmathematics, research scholars, and professors from localcolleges. It was planned to celebrate the 125th birthdayin a grand manner by inviting the foreign eminent math-ematical scholars of this century viz., G E Andrews. andBruce C Berndt, who are very familiar with the contribu-tions and works of Ramanujan.Ramanujan’s work and life are celebrated on 22 Decem-ber at the Indian Institute of Technology (IIT), Madrasin Chennai. The Department of Mathematics celebratesthis day by organising a National Symposium on Math-ematical Methods and Applications (NSMMA) for oneday by inviting eminent Indian and foreign scholars.A prize for young mathematicians from developingcountries has been created in the name of Ramanu-jan by the International Centre for Theoretical Physics(ICTP), in co-operation with the International Mathe-matical Union, which nominate members of the prizecommittee. The Shanmugha Arts, Science, Technology& Research Academy (SASTRA), based in the state ofTamil Nadu in South India, has instituted the SASTRARamanujan Prize of $10,000 to be given annually to amathematician not exceeding the age of 32 for outstand-ing contributions in an area of mathematics influenced byRamanujan. The age limit refers to the years Ramanu-jan lived, having nevertheless still achieved many accom-plishments. This prize has been awarded annually since2005, at an international conference conducted by SAS-TRA in Kumbakonam, Ramanujan’s hometown, aroundRamanujan’s birthday, 22 December.On the 125th anniversary of his birth, India declaredthe birthday of Ramanujan, 22 December, as 'NationalMathematics Day.' The declaration was made by Dr.Manmohan Singh in Chennai on 26 December 2011.[98]Dr Manmohan Singh also declared that the year 2012would be celebrated as the National Mathematics Year.His residence is now preserved by SASTRA university inKumbakonam.

8 In popular culture

• Ramanujan, an Indo-British collaboration film,chronicling the life of Ramanujan, is being madeby the independent film company Camphor Cin-ema.[99] The cast and crew include director GnanaRajasekaran, cinematographer Sunny Joseph andeditor B. Lenin.[100][101] Popular Indian and Englishstars Abhinay Vaddi, Suhasini Maniratnam, Bhama,Kevin McGowan and Michael Lieber star in pivotalroles.[102]

9

• Ramanujan is referenced in the 1997 American filmGood Will Hunting.

• A film, based on the book The Man Who KnewInfinity: A Life of the Genius Ramanujan byRobert Kanigel, is being made by Edward Press-man andMatthew Brown with R. Madhavan playingRamanujan.[103]

• A play, First Class Man by Alter EgoProductions,[104] was based on David Free-man’s First Class Man. The play is centred aroundRamanujan and his complex and dysfunctionalrelationship with Hardy. On 16 October 2011,it was announced that Roger Spottiswoode, bestknown for his James Bond film Tomorrow NeverDies, is working on the film version, starring actorSiddharth. Like the book and play it is also titledThe First Class Man.[105]

• ADisappearing Number is a recent British stage pro-duction by the companyComplicite that explores therelationship between Hardy and Ramanujan.

• The novel The Indian Clerk by David Leavitt ex-plores in fiction the events following Ramanujan’sletter to Hardy.[106][107]

• On 22 March 1988, the PBS Series Nova aireda documentary about Ramanujan, “The Man WhoLoved Numbers” (Season 15, Episode 19).[108]

• Google honoured him on his 125th birth anniver-sary by replacing its logo with a doodle on its homepage.[109]

• The television series Numb3rs has the character Dr.Amita Ramanujan, a professor of applied mathe-matics, named after Ramanujan[110]

• Ramanujan’s story is both referenced and echoed inCyril M. Kornbluth's “Gomez”.

9 See also

• List of amateur mathematicians

• Ramanujan graph

• Ramanujan summation

• Ramanujan’s constant

• Ramanujan’s ternary quadratic form

• Rank of a partition

• 2719 (number)

• List of Indian mathematicians

10 Notes[1] C.P. Snow Foreword to "A Mathematician’s Apology" by

G.H. Hardy

[2] Berndt, Bruce C. (2005). Ramanujan’s Notebooks Part V.SpringerLink. p. 4. ISBN 0-387-94941-0.

[3] “Rediscovering Ramanujan”. Frontline 16 (17): 650. Au-gust 1999. Retrieved 20 December 2012.

[4] Ono, Ken (June–July 2006). “Honoring a Gift fromKum-bakonam” (PDF). Notices of the American MathematicalSociety (Mathematical Association of America) 53 (6):650. Retrieved 23 June 2007.

[5] Alladi, Krishnaswami (1998). Analytic and ElementaryNumber Theory: A Tribute to Mathematical Legend PaulErdös. Norwell, Massachusetts: Kluwer Academic Pub-lishers. p. 6. ISBN 0-7923-8273-0.

[6] Kanigel 1991, p. 11

[7] Kanigel 1991, pp. 17–18

[8] Berndt & Rankin 2001, p. 89

[9] Kanigel 1991, p. 12

[10] Kanigel 1991, p. 13

[11] Kanigel 1991, p. 19

[12] Kanigel 1991, p. 14

[13] Kanigel 1991, p. 20

[14] Kanigel 1991, p. 25

[15] Berndt & Rankin 2001, p. 9

[16] Hardy, G. H. (1999). Ramanujan: Twelve Lectures onSubjects Suggested by His Life and Work. Providence,Rhode Island: American Mathematical Society. p. 2.ISBN 0-8218-2023-0.

[17] Kanigel 1991, p. 27

[18] Kanigel 1991, p. 39

[19] A to Z of mathematicians by Tucker McElroy 2005 ISBN0-8160-5338-3-page 221

[20] Collected papers of Srinivasa Ramanujan Srinivasa Ra-manujan Aiyangar, Godfrey Harold Hardy, P. Veṅkates-vara Seshu Aiyar 2000 ISBN 0-8218-2076-1 page xii

[21] Kanigel 1991, p. 90

[22] Kanigel 1991, p. 28

[23] Kanigel 1991, p. 45

[24] Kanigel 1991, p. 47

[25] “Ramanujan lost and found: a 1905 letter from TheHindu". The Hindu (Chennai, India). 25 December 2011.

[26] Kanigel 1991, pp. 48–49

[27] Kanigel 1991, pp. 55–56

10 10 NOTES

[28] Kanigel 1991, p. 71

[29] “Ramanujan’s wife: Janakiammal (Janaki)". Institute ofMathematical Sciences, Chennai. Retrieved 10 Novem-ber 2012.

[30] Kanigel 1991, p. 72

[31] Ramanujan, Srinivasa (1968). P. K. Srinivasan, ed. Ra-manujan Memorial Number: Letters and Reminiscences.Madras: Muthialpet High School. Vol. 1, p100.

[32] Kanigel 1991, p. 73

[33] Kanigel 1991, pp. 74–75

[34] Ranganathan, Shiyali Ramamrita (1967). Ramanujan:The Man and the Mathematician. Bombay: Asia Publish-ing House., p. 23.

[35] Srinivasan (1968), Vol. 1, p99.

[36] Kanigel 1991, p. 77

[37] Srinivasan (1968), Vol. 1, p129.

[38] Srinivasan (1968), Vol. 1, p86.

[39] Neville, Eric Harold (January 1921). “The Late Srini-vasa Ramanujan”. Nature 106 (2673): 661–662.Bibcode:1921Natur.106..661N. doi:10.1038/106661b0.

[40] Ranganathan 1967, p. 24

[41] Kanigel 1991, p. 80

[42] Kanigel 1991, p. 86

[43] Kanigel 1991, p. 87

[44] Kanigel 1991, p. 91

[45] Seshu Iyer, P. V. (June 1920). “The Late Mr. S. Ramanu-jan, B.A., F.R.S”. Journal of the Indian Mathematical So-ciety 12 (3): 83.

[46] Neville (March 1942), p292.

[47] Srinivasan (1968), p176.

[48] Srinivasan (1968), p31.

[49] Srinivasan (1968), p49.

[50] Kanigel 1991, p. 96

[51] Kanigel 1991, p. 105

[52] Letter from M. J. M. Hill to a C. L. T. Griffith (a for-mer student who sent the request to Hill on Ramanujan’sbehalf), 28 November 1912.

[53] Kanigel 1991, p. 106

[54] Kanigel 1991, pp. 170–171

[55] Snow, C. P. (1966). Variety of Men. New York: CharlesScribner’s Sons. pp. 30–31.

[56] Hardy, G. H. (June 1920). “Obituary, S. Ramanujan”.Nature 105 (7): 494. Bibcode:1920Natur.105..494H.doi:10.1038/105494a0.

[57] Kanigel 1991, p. 167

[58] Kanigel 1991, p. 168

[59] Hardy (June 1920), pp494–495.

[60] Neville, Eric Harold (March 1942). “Srini-vasa Ramanujan”. Nature 149 (3776): 293.Bibcode:1942Natur.149..292N. doi:10.1038/149292a0.

[61] Letter, Hardy to Ramanujan, 8 February 1913.

[62] Letter, Ramanujan to Hardy, 22 January 1914.

[63] Kanigel 1991, p. 185

[64] Letter, Ramanujan to Hardy, 27 February 1913,Cambridge University Library.

[65] Kanigel 1991, p. 175

[66] Ram, Suresh (1972). Srinivasa Ramanujan. New Delhi:National Book Trust. p. 29.

[67] Ranganathan 1967, pp. 30–31

[68] Ranganathan 1967, p. 12

[69] Kanigel 1991, p. 183

[70] Kanigel 1991, p. 184

[71] Kanigel 1991, p. 196

[72] Kanigel 1991, p. 202

[73] Hardy, G. H. (1940). Ramanujan. Cambridge:Cambridge University Press. p. 10.

[74] Letter, Littlewood to Hardy, early March 1913.

[75] Hardy, G. H. (1979). Collected Papers of G. H. Hardy.Oxford, England: Clarendon Press. Vol. 7, p720.

[76] Kanigel 1991, pp. 299–300

[77] Peterson, Doug. “Raiders of the Lost Notebook”. UIUCCollege of Liberal Arts and Sciences. Retrieved 11 Jan-uary 2014.

[78] “Ramanujan’s Personality”.

[79] Kanigel 1991, p. 36

[80] Kanigel 1991, p. 281

[81] “Quote by Srinivasa Ramanujan Iyengar”.

[82] Chaitin, Gregory (28 July 2007). “Less Proof, MoreTruth”. NewScientist (2614): 49.

[83] Kanigel 1991, p. 283

[84] Ranganathan 1967, p. 82

[85] Calyampudi Radhakrishna Rao (1997). Statistics andtruth: putting chance to work. World Scientific. p. 185.ISBN 978-981-02-3111-8. Retrieved 7 June 2010.

[86] “Partition Formula”.

11

[87] “100-Year-Old Deathbed Dreams of MathematicianProved True”. Fox News. 28 December 2012.

[88] Ono (June–July 2006), p649.

[89] “Ramanujans Notebooks”.

[90] “Quotations by Hardy”. Gap.dcs.st-and.ac.uk. Retrieved20 November 2012.

[91] “Ramanujan quote”.

[92] Srinivasa Ramanujan. Retrieved 2 December 2010.

[93] K Srinivasa Rao. “Srinivasa Ramanujan (22 December1887 – 26 April 1920)".

[94] “Bruce Berndt on “Ramanujan’s Lost Notebook”, IITMadras, 24th May 2011”. youtube.com.

[95] “Stamps released in 1962”. Indian Postage Stamps. Re-trieved 22 May 2012.

[96] “Stamps 2011”. India Post. Retrieved 22 May 2012.

[97] “India Post Issued a Commemorative Stamp on S Ra-manujan”. Phila Mirror. 26 December 2011. Retrieved22 May 2012.

[98] “News / National :". CNN IBN. India. Retrieved 26 De-cember 2011.

[99] "'Ramanujan' Makers Shoot in His House”. Indiatimes(Times Internet Limited.). Retrieved 12 July 2013.

[100] “Camphor Cinema Presents Their First FilmRamanujan”.Box Office India. Select Publishing Company. Retrieved12 July 2013.

[101] “Makers of 'Ramanujan' shoot in genius’ house”. http://zeenews.india.com/. Zee Media Corporation Ltd. Re-trieved 12 July 2013.

[102] Krishnamachari, Suganthy (27 June 2013). “Travails of agenius”. The Hindu (Chennai, India). Retrieved 12 July2013.

[103] “Two Hollywood movies on Ramanujan soon”. Sify.com.30 March 2006. Retrieved 24 July 2014.

[104] “First Class Man”. Alteregoproductions.org. Retrieved20 November 2012.

[105] “News / National : James Bond director to make film onRamanujan”. The Hindu (India). 16 October 2011. Re-trieved 18 October 2011.

[106] Nell Freudenberger (16 September 2007). “Lust forNumbers”. The New York Times. Retrieved 4 Septem-ber 2011.

[107] DJ Taylor (26 January 2008). “Adding up to a life”. TheGuardian (UK). Retrieved 4 September 2011.

[108] “The Man Who Loved Numbers”. Pbs.org. Retrieved 18October 2011.

[109] “Google doodles for Ramanujan’s 125th birthday”. Timesof India. 22 December 2012. Archived from the originalon 22 December 2012. Retrieved 22 December 2012.

[110] http://www.tv.com/people/navi-rawat/

11 Selected publications by Ra-manujan

• Srinivasa Ramanujan, G. H. Hardy, P. V. Seshu Ai-yar, B. M. Wilson, Bruce C. Berndt (2000). Col-lected Papers of Srinivasa Ramanujan. AMS. ISBN0-8218-2076-1.

This book was originally published in 1927 af-ter Ramanujan’s death. It contains the 37 pa-pers published in professional journals by Ra-manujan during his lifetime. The third reprintcontains additional commentary by Bruce C.Berndt.

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

These books contain photocopies of the origi-nal notebooks as written by Ramanujan.

• S. Ramanujan (1988). The Lost Notebook and OtherUnpublished Papers. New Delhi: Narosa. ISBN 3-540-18726-X.

This book contains photo copies of the pagesof the “Lost Notebook”.

• Problems posed by Ramanujan, Journal of the In-dian Mathematical Society.

• S. Ramanujan (2012). Notebooks (2 Volumes).Bombay: Tata Institute of Fundamental Research.

This was produced from scanned and micro-filmed images of the original manuscripts byexpert archivists of RojaMuthiah Research Li-brary, Chennai.

12 Selected publications about Ra-manujan and his work

• Berndt, Bruce C. (1998). Butzer, P. L.; Oberschelp,W.; Jongen, H. Th., ed. Charlemagne and His Her-itage: 1200 Years of Civilization and Science in Eu-rope (PDF). Turnhout, Belgium: Brepols Verlag.pp. 119–146. ISBN 2-503-50673-9.

• Berndt, Bruce C.; Andrews, George E. (2005).Ramanujan’s Lost Notebook. Part I. New York:Springer. ISBN 0-387-25529-X.

• Berndt, Bruce C.; Andrews, George E. (2008).Ramanujan’s Lost Notebook. Part II. New York:Springer. ISBN 978-0-387-77765-8.

12 13 EXTERNAL LINKS

• Berndt, Bruce C.; Andrews, George E. (2012).Ramanujan’s Lost Notebook. Part III. New York:Springer. ISBN 978-1-4614-3809-0.

• Berndt, Bruce C.; Andrews, George E. (2013).Ramanujan’s Lost Notebook. Part IV. New York:Springer. ISBN 978-1-4614-4080-2.

• Berndt, Bruce C.; Rankin, Robert A. (1995). Ra-manujan: Letters and Commentary 9. Providence,Rhode Island: American Mathematical Society.ISBN 0-8218-0287-9.

• Berndt, Bruce C.; Rankin, Robert A. (2001). Ra-manujan: Essays and Surveys 22. Providence,Rhode Island: American Mathematical Society.ISBN 0-8218-2624-7.

• Berndt, Bruce C. (2006). Number Theory in theSpirit of Ramanujan 9. Providence, Rhode Island:American Mathematical Society. ISBN 0-8218-4178-5.

• Berndt, Bruce C. (1985). Ramanujan’s Notebooks.Part I. New York: Springer. ISBN 0-387-96110-0.

• Berndt, Bruce C. (1999). Ramanujan’s Notebooks.Part II. New York: Springer. ISBN 0-387-96794-X.

• Berndt, Bruce C. (2004). Ramanujan’s Notebooks.Part III. New York: Springer. ISBN 0-387-97503-9.

• Berndt, Bruce C. (1993). Ramanujan’s Notebooks.Part IV. New York: Springer. ISBN 0-387-94109-6.

• Berndt, Bruce C. (2005). Ramanujan’s Notebooks.Part V. New York: Springer. ISBN 0-387-94941-0.

• Hardy, G. H. (1978). Ramanujan. New York:Chelsea Pub. Co. ISBN 0-8284-0136-5.

• Hardy, G. H. (1999). Ramanujan: Twelve Lectureson Subjects Suggested by His Life and Work. Prov-idence, Rhode Island: American Mathematical So-ciety. ISBN 0-8218-2023-0.

• Henderson, Harry (1995). Modern Mathematicians.New York: Facts on File Inc. ISBN 0-8160-3235-1.

• Kanigel, Robert (1991). The Man Who Knew In-finity: a Life of the Genius Ramanujan. New York:Charles Scribner’s Sons. ISBN 0-684-19259-4.

• Kolata, Gina (19 June 1987). “Remember-ing a 'Magical Genius’". Science, New Series(American Association for the Advance-ment of Science) 236 (4808): 1519–1521.doi:10.1126/science.236.4808.1519.

• Leavitt, David (2007). The Indian Clerk (paperbacked.). London: Bloomsbury. ISBN 978-0-7475-9370-6.

• Narlikar, Jayant V. (2003). Scientific Edge: the In-dian Scientist From Vedic to Modern Times. NewDelhi, India: Penguin Books. ISBN 0-14-303028-0.

• Sankaran, T. M. (2005). “Srinivasa Ramanujan-Ganitha lokathile Mahaprathibha” (in Malayalam).Kochi, India: Kerala Sastra Sahithya Parishath.

13 External links

13.1 Media links

• Biswas, Soutik (16 March 2006). “Film to celebratemathematics genius”. BBC. Retrieved 24 August2006.

• Feature Film onMathematics Genius Ramanujan byDev Benegal and Stephen Fry

• BBC radio programme about Ramanujan – episode5

• A biographical song about Ramanujan’s life

• P.B.S. Nova Series: “The Man Who Loved Num-bers” (1988)

13.2 Biographical links

• Srinivasa Ramanujan at the Mathematics GenealogyProject

• O'Connor, John J.; Robertson, Edmund F.,“Srinivasa Ramanujan”, MacTutor History ofMathematics archive, University of St Andrews.

• Weisstein, Eric W., Ramanujan, Srinivasa (1887–1920) from ScienceWorld.

• Srinivasa Aiyangar Ramanujan

• A short biography of Ramanujan

• “Our Devoted Site for Great Mathematical Genius”

13.3 Other links

• A Study Group For Mathematics: Srinivasa Ra-manujan Iyengar

• The Ramanujan Journal – An international journaldevoted to Ramanujan

• International Math Union Prizes, including a Ra-manujan Prize.

13.3 Other links 13

• Hindu.com: Norwegian and Indian mathematicalgeniuses, RAMANUJAN – Essays and Surveys,Ramanujan’s growing influence, Ramanujan’s men-tor

• Hindu.com: The sponsor of Ramanujan

• Bruce C. Berndt; Robert A. Rankin (2000).“The Books Studied by Ramanujan in India”.American Mathematical Monthly (MathematicalAssociation of America) 107 (7): 595–601.doi:10.2307/2589114. JSTOR 2589114. MR1786233.

• “Ramanujan’s mock theta function puzzle solved”

• Ramanujan’s papers and notebooks

• Sample page from the second notebook

• Ramanujan on Fried Eye

• Clark, Alex. “163 and Ramanujan Constant”. Num-berphile. Brady Haran.

14 14 TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES

14 Text and image sources, contributors, and licenses

14.1 Text• Srinivasa Ramanujan Source: http://en.wikipedia.org/wiki/Srinivasa%20Ramanujan?oldid=637906457 Contributors: AxelBoldt, TheAnome, Jeronimo, XJaM, William Avery, AdamRetchless, Pichai Asokan, Michael Hardy, Shyamal, Gabbe, Menchi, Chinju, Lquilter,GTBacchus, Mcarling, Paddu, Card, Ahoerstemeier, Ijon, Sray, Poor Yorick, Jengod, JidGom, Charles Matthews, NilsB, Dfeuer, Nataraja,Jitse Niesen, Zoicon5, Tpbradbury, Ldrhcp, Furrykef, Nv8200p, Taxman, VeryVerily, Phys, Topbanana, AnonMoos, Eugene van derPijll, Chuunen Baka, Robbot, Vincent Gray, Sander123, Jaredwf, Fredrik, Chocolateboy, Lowellian, DHN, Jondel, Andrew Levine, Rr-janbiah, JackofOz, Ambarish, Anthony, Superm401, MikeCapone, Ancheta Wis, Exploding Boy, Giftlite, Nichalp, Vishvas vasuki, ÆvarArnfjörð Bjarmason, Lethe, Berasategui, Peruvianllama, Everyking, Gro-Tsen, Curps, Gamaliel, Jason Quinn, Neilc, Etaonish, Zarvok,Iceager, DragonflySixtyseven, Thincat, Jawed, Icairns, Urhixidur, Ukexpat, Robin klein, Karl Dickman, Abdull, Jason Carreiro, Sha-hab, Venu62, Indosauros, Pyrop, RossPatterson, Discospinster, Rich Farmbrough, Guanabot, FT2, Vsmith, Arthur Holland, Ashwatham,Paul August, Bender235, Billlion, CanisRufus, Alren, Pt, VishalB, Szquirrel, RoyBoy, AlvinMGO, Gershwinrb, Causa sui, Bobo192,Evolauxia, Blakkandekka, Arcadian, QTxVi4bEMRbrNqOorWBV, La goutte de pluie, ריינהארט ,לערי Larry V, LostLeviathan, He-lix84, Haham hanuka, JesseHogan, Merope, Ranveig, Jumbuck, Alansohn, Arthena, Salilb, Riana, SlimVirgin, Gaurav1146, Mrholybrain,NTK, LavosBacons, Wtmitchell, Velella, Hasdrubal, VivaEmilyDavies, Staeiou, Vadakkan, Rajprem, Oleg Alexandrov, Zntrip, Velho,Jak86, Woohookitty, Linas, Mindmatrix, Shreevatsa, Malapati, Pol098, Dodiad, Julien Tuerlinckx, Mpatel, Slocombe, Code Wizard,Nandakumarg, Markizs, GregorB, Vanished user 05, Isnow, Waldir, Liface, DESiegel, Jan.bannister, Raguks, Graham87, Deltabeignet,Pranathi, Dwaipayanc, Viswaprabha, Rjwilmsi, Pdelong, Koavf, Zbxgscqf, Srichrome, Bruce1ee, TheRingess, Tawker, Mike Peel, Don-Siano, R.e.b., Sango123, Aveekbh, Fish and karate, Baryonic Being, Mtommila, Da Stressor, Maxal, Pete.Hurd, Alphachimp, Themiss-inglint, Glenn L, Gareth E Kegg, IntrepidWill, Haonhien, Chobot, DaGizza, Jagdeep, Adoniscik, EamonnPKeane, YurikBot, Wavelength,RobotE, Ventolin, Conscious, Zafiroblue05, Lenthe, Thoreaulylazy, Gaius Cornelius, Member, NawlinWiki, Rak3sh, Shreshth91, That-dog, Lowe4091, Chick Bowen, Banes, DYLAN LENNON, Moe Epsilon, Mikeblas, Hv, Jalabi99, Nick C, Tony1, Zwobot, Supten,Samir, Gadget850, DeadEyeArrow, Bota47, Haemo, Black Falcon, Gnusbiz, Wknight94, Ms2ger, Crisco 1492, Zargulon, Ario, PaulMagnussen, Ninly, Bhumiya, Djkimmons, Arthur Rubin, Sarefo, Tevildo, LeonardoRob0t, Shyam, Arundhati bakshi, Pred, Garion96,Ilmari Karonen, Chitrada, Katieh5584, Kungfuadam, Banus, Cyphase, DVD R W, Vineethtm, Sintonak.X, GrafZahl, SmackBot, Drum-mondjacob, YellowMonkey, Roger Hui, Lestrade, Herostratus, InverseHypercube, Pgk, Jagged 85, Thunderboltz, Eskimbot, Dexter73,Alsandro, SmartGuy Old, Msrkiran, Commander Keane bot, Eiler7, Xaosflux, Gilliam, Andy M. Wang, Rst20xx, Bvssatish, Pseinstein,Vignesh.ks, Zouf, Bluebot, Droll, Acid2base, TheFeds, Chisophugis, Exitr, Nbarth, Whispering, Colonies Chris, Lenin and McCarthy,OrphanBot, Onorem, Vanished User 0001, Gala.martin, LouScheffer, Cribananda, Hypergeometric2F1(a,b,c,x), Robma, Fuhghettaboutit,Jackohare, DRLB, Allansteel, G716, Kshieh, N Shar, Pilotguy, Vaibhavgarg, Ohconfucius, RNLion, Lambiam, Nishkid64, ArglebargleIV,Wideangle, Ser Amantio di Nicolao, Gloriamarie, Harryboyles, Mouse Nightshirt, Saravan p, Rohit math, Titus III, Richard L. 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14.2 Images 15

Magentic Manifestations, Webclient101, Nandagopal.v, Aumkaar Pranav, Aloak1, Siberian Patriot, Nikhilvsrockz, Frosty, Graphium,HullIntegrity, Danny Sprinkle, Ranjyotiprakashsingh, Dravidianhero, PinkAmpersand, TamBram, Epicgenius, Missionedit, Piesiva, Irahul-pandey, Abhiraj m, Jamesmcmahon0, Eshwar.om, Sosthenes12, Jakec, EvergreenFir, Gouthams2, Timpu454, Whitefluffyduck, RajashreeRajaram, The Herald, AioftheStorm, Ginsuloft, Pratik.patra, Kartik61182, Varun.a8, Aryan203, ShivamPatel scientist, Differentthought-tomaths, Monkbot, Akpelirw, Skwwtlu, Alex20141729, Ice ax1940ice pick, Karyakarta, Sanjeev2015, TanmayDaga, Vijayjeyasanksr,Haywardcir, SSoheilHosseini, D.v.karthick, Vivek Sarje and Anonymous: 946

14.2 Images• File:Commons-logo.svg Source: http://upload.wikimedia.org/wikipedia/en/4/4a/Commons-logo.svg License: ? Contributors: ? Originalartist: ?

• File:RamanujanCambridge.jpg Source: http://upload.wikimedia.org/wikipedia/commons/5/57/RamanujanCambridge.jpg License:Public domain Contributors: http://www.educ.fc.ul.pt/docentes/opombo/seminario/Ramanujan/Ramanujan22.jpg Original artist: CharlesF. Wilson

• File:Ramanujanhome.jpg Source: http://upload.wikimedia.org/wikipedia/commons/3/31/Ramanujanhome.jpg License: Public domainContributors: Transferred from en.wikipedia to Commons. Original artist: Adiswini at English Wikipedia

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• File:Srinivasa_Ramanujam_bust_BITM.JPG Source: http://upload.wikimedia.org/wikipedia/commons/d/d3/Srinivasa_Ramanujam_bust_BITM.JPG License: CC-BY-SA-3.0 Contributors: Own work Original artist: AshLin

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