-
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/sri.ni.va.sra.ma.n.dn/)
(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,
innite series,and continued fractions. Ramanujan initially
developedhis own mathematical research in isolation; it wasquickly
recognized by Indian mathematicians. Whenhis skills became apparent
to the wider mathematicalcommunity, centred 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.During his short life, Ramanujan independently
compilednearly 3900 results (mostly identities and
equations).[1]Nearly all his claims have now been proven correct,
al-though a small number of these results were actually falseand
some were already known.[2] 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.[3]The Ramanujan Journal, an international
publication,was launched to publish work in all areas of
mathematicsinuenced by his work.[4]
1 Early lifeRamanujan was born on 22 December 1887 in
Erode,Madras Presidency (now Pallipalayam, Erode, TamilNadu), at
the residence of his maternal grandparents.[5]His father, K.
Srinivasa Iyengar, worked as a clerk in asari shop and hailed from
the district of Thanjavur.[6] Hismother, Komalatammal, was a
housewife and also sangat a local temple.[7] 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,unlike thousands in the Thanjavur District who died
fromthe disease that year.[8] He moved with his mother to
Ramanujans home on Sarangapani Street, Kumbakonam
her 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.[9] In March 1894, he
was moved to a Tamilmedium school. After his maternal grandfather
lost hisjob as a court ocial in Kanchipuram,[10] Ramanujanand his
mother moved back to Kumbakonam and he wasenrolled in the Kangayan
Primary School.[11] 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.[11]
Since Ramanujans 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,to attend pujas at the temple and particular
eating habits all of which are part of Brahmin culture.[12] At
the
1
-
2 2 ADULTHOOD IN INDIA
Kangayan 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 rst in thedistrict.[13] That year, Ramanujan entered Town
HigherSecondary School where he encountered formal mathe-matics for
the rst time.[13]
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.[14][15] He
completely mastered this book bythe age of 13 and discovered
sophisticated theorems onhis own. By 14, he was receiving merit
certicates 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.[16] He completed mathematical ex-ams in half the
allotted time, and showed a familiaritywith geometry and innite
series. Ramanujan was shownhow to solve cubic equations in 1902 and
he went on tond 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.[17][18] 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.[19] The book
is gen-erally acknowledged as a key element in awakening thegenius
of Ramanujan.[19] The next year, he had indepen-dently developed
and investigated the Bernoulli numbersand had calculated the
EulerMascheroni constant up to15 decimal places.[20] His peers at
the time commentedthat they rarely understood him and stood in
respectfulawe of him.[16]
When he graduated from Town Higher Secondary Schoolin 1904,
Ramanujan was awarded the K. RanganathaRao prize for mathematics by
the schools headmas-ter, Krishnaswami Iyer. Iyer introduced
Ramanujanas an outstanding student who deserved scores higherthan
the maximum possible marks.[16] He received ascholarship to study
at Government Arts College, Kum-bakonam,[21][22] 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.[23] In August 1905, he ran awayfrom home, heading
towards Visakhapatnam and stayedin Rajahmundry[24] for about a
month.[25] He later en-rolled at Pachaiyappas 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 on
the brink of starvation.[26]
2 Adulthood in IndiaOn 14 July 1909, Ramanujan was married to a
ten-year-old bride, Janakiammal (21 March 1899 13 April1994).[27]
She came from Rajendram, a village close toMarudur (Karur district)
Railway Station. Ramanujansfather did not participate in the
marriage ceremony.[28]
After the marriage, Ramanujan developed a hydroceletestis, an
abnormal swelling of the tunica vaginalis, an in-ternal membrane in
the testicle.[29] The condition couldbe treated with a routine
surgical operation that wouldrelease the blocked uid 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.[30]
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.[31]
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, tohand these [Ramanujans
mathematical notebooks] overto Professor Singaravelu Mudaliar [the
mathematics pro-fessor at Pachaiyappas College] or to the British
profes-sor Edward B. Ross, of theMadras Christian College.[32]After
Ramanujan recovered and got back his notebooksfrom Iyer, he took a
northbound train fromKumbakonamto Villupuram, a coastal city under
French control.[33][34]
2.1 Attention towards mathematics
Ramanujan met deputy collector V. Ramaswamy Aiyer,who had
recently founded the Indian MathematicalSociety.[35] 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.[36]
Ramaswamy Aiyer sent Ramanujan, with letters of in-troduction,
to his mathematician friends in Madras.[35]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 IndianMathematical Society.[37][38][39]
Ramachandra Rao was
-
2.2 Contacting English mathematicians 3
impressed by Ramanujans 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.[40] Ramanujans friend, C. V.
Ra-jagopalachari, persisted with Ramachandra Rao and triedto quell
any doubts over Ramanujans 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
Ramanujansmathematical brilliance.[40] When Rao asked him whathe
wanted, Ramanujan replied that he needed some workand nancial
support. Rao consented and sent him toMadras. He continued his
mathematical research withRaos nancial 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.[41]
One of the rst problems he posed in the journal was:
r1 + 2
q1 + 3
p1 + :
He waited for a solution to be oered 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 rst
notebook, he formulated an equation thatcould be used to solve the
innitely nested radicals prob-lem.
x+n+a =
rax+ (n+ a)2 + x
qa(x+ n) + (n+ a)2 + (x+ n)
p
Using this equation, the answer to the question posedin the
Journal was simply 3.[42] Ramanujan wrote hisrst 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 Bnn 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
BernoullisNumbers, Ramanujan gave three proofs, two corollariesand
three conjectures.[43] Ramanujans writing initiallyhad many aws. As
Journal editor M. T. Narayana Iyen-gar noted:
Mr. Ramanujans 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.[44]
Ramanujan later wrote another paper and also continuedto provide
problems in the Journal.[45] In early 1912, hegot a temporary job
in the Madras Accountant General'soce, with a salary of 20 rupees
per month. He lasted foronly a few weeks.[46] 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 oce, 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 quitecondent
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.[47]
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.[48] 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.[49] At his oce,
Ramanujan easily andquickly completed the work he was given, so he
spent hisspare time doing mathematical research. Ramanujansboss,
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 mathematiciansIn the spring of 1913,
Narayana Iyer, Ramachandra Raoand E.W.Middlemast tried to present
Ramanujans workto British mathematicians. One mathematician, M. J.
M.Hill of University College London, commented that Ra-manujans
papers were riddled with holes.[50] He said thatalthough Ramanujan
had a taste for mathematics, andsome ability, he lacked the
educational background andfoundation needed to be accepted by
mathematicians.[51]Although Hill did not oer 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.[52]
The rst two professors, H. F. Baker and E. W. Hob-son, returned
Ramanujans papers without comment.[53]On 16 January 1913, Ramanujan
wrote to G. H. Hardy.Coming from an unknown mathematician, the nine
pagesof mathematics made Hardy initially view Ramanu-jans
manuscripts as a possible fraud.[54] Hardy recog-nised some of
Ramanujans formulae but others seemedscarcely possible to
believe.[55] One of the theoremsHardy found scarcely possible to
believe was on the bot-tom of page three (valid for 0 < a < b
+ 1/2):
Z 10
1 + x2/(b+ 1)2
1 + x2/(a)2 1 + x
2/(b+ 2)2
1 + x2/(a+ 1)2 dx =
p
2(a+
12 )(b+ 1)(b a+ 12 )
(a)(b+ 12 )(b a+ 1):
Hardy was also impressed by some of Ramanujans otherwork
relating to innite series:
151
2
3+9
1 32 4
313
1 3 52 4 6
3+ = 2
1+9
1
4
4+17
1 54 8
4+25
1 5 94 8 12
4+ = 2
32
122
34
:The rst 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 rst been
researchedby Leonhard Euler and Carl Friedrich Gauss. Comparedto
Ramanujans work on integrals, Hardy found these re-sults much more
intriguing.[56] After he saw Ramanu-jans 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.[57] He gured thatRamanujans theorems
must be true, because, if theywere not true, no one would have the
imagination to in-vent them.[57] 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 receivedand commented that Ramanujan was
a mathematicianof the highest quality, a man of altogether
exceptionaloriginality and power.[58] One colleague, E. H.
Neville,later commented that not one [theorem] could have beenset
in the most advanced mathematical examination in theworld.[59]
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.[60] Before his
letter arrived in Madrasduring the third week of February, Hardy
contacted theIndian Oce to plan for Ramanujans trip to
Cambridge.Secretary Arthur Davies of the Advisory Committee for
Indian Students met with Ramanujan to discuss the over-seas
trip.[61] In accordance with his Brahmin upbring-ing, Ramanujan
refused to leave his country to go toa foreign land.[62] Meanwhile,
Ramanujan sent a letterpacked with theorems to Hardy, writing, I
have found afriend in you who views my labour
sympathetically.[63]
To supplement Hardys endorsement, a former mathe-matical
lecturer at Trinity College, Cambridge, GilbertWalker, looked at
Ramanujans work and expressedamazement, urging him to spend time at
Cambridge.[64]As a result of Walkers endorsement, B. HanumanthaRao,
a mathematics professor at an engineering college,invited
Ramanujans colleague Narayana Iyer to a meet-ing of the Board of
Studies in Mathematics to discusswhat we can do for S.
Ramanujan.[65] The board agreedto grant Ramanujan a research
scholarship of 75 ru-pees per month for the next two years at the
Universityof Madras.[66] 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 days mail.[67] In his quarterly pa-pers,
Ramanujan drew up theorems to make denite in-tegrals more easily
solvable. Working o Giuliano Frul-lanis 1821 integral theorem,
Ramanujan formulated gen-eralisations that could be made to
evaluate formerly un-yielding integrals.[68]
Hardys 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.[69]
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.[59] Apparently,Ramanujans 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 fullment of
hislifes purpose.[59] Ramanujan then set sail for England,leaving
his wife to stay with his parents in India.
3 Life in EnglandRamanujan boarded the S.S. Nevasa on 17 March
1914,and at 10 o'clock in the morning, the ship departed
fromMadras.[70] 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
Whewells 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 Nevilles house and took up
residenceonWhewells Court, just a ve-minute walk
fromHardysroom.[71] Hardy and Ramanujan began to take a look
atRamanujans notebooks. Hardy had already received 120theorems from
Ramanujan in the rst 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.[72] Ramanujan left a deep impres-sion on Hardy and
Littlewood. Littlewood commented,I can believe that hes at least a
Jacobi",[73] while Hardysaid he can compare him only with
[Leonhard] Euler orJacobi.[74]
Ramanujan spent nearly ve years in Cambridge collab-orating with
Hardy and Littlewood and published a partof his ndings there. Hardy
and Ramanujan had highlycontrasting personalities. Their
collaboration was a clashof dierent 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 ll the gaps in Ramanujanseducation 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 rstpart of which was published as a paper in the Proceed-ings
of the London Mathematical Society. The paper wasover 50 pages with
dierent 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 rst Indian to be elected a Fellow ofTrinity College,
Cambridge.[75]
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, Ramanujans 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
vitamindeciency and was conned 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.[28]
A 1994 analysis of Ramanujans 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,[76] a dicult disease to diagnose, but
oncediagnosed readily cured.[76]
-
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 digniedmanwith pleas-ant manners.[77] He
lived a rather spartan life while atCambridge. Ramanujans rst
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,[78] 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.[79] He often
said,An equation for me has no meaning, unless it representsa
thought of God.[80][81]
Hardy cites Ramanujan as remarking that all religionsseemed
equally true to him.[82] Hardy further arguedthat Ramanujans
religiousness had been romanticised byWesterners and overstatedin
reference to his belief, notpracticeby Indian biographers. At the
same time, heremarked on Ramanujans strict observance of
vegetari-anism.
4 Mathematical achievementsIn mathematics, there is a
distinction between having aninsight and having a proof. Ramanujans
talent suggesteda plethora of formulae that could then be
investigated indepth later. It is said by G. H. Hardy that
Ramanujansdiscoveries 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 intriguinginnite series
for , one of which is given below
1
=
2p2
9801
1Xk=0
(4k)!(1103 + 26390k)
(k!)43964k:
This result is based on the negative fundamental discrim-inant d
= 458 = 232 with class number h(d) = 2(note that 571358 = 26390 and
that 9801=9999;396=499) and is related to the fact that
ep58 = 3964 104:000000177 : : : :
Compare to Heegner numbers, which have class num-ber 1 and yield
similar formulae. Ramanujans series for converges extraordinarily
rapidly (exponentially) andforms the basis of some of the fastest
algorithms currentlyused to calculate . Truncating the sum to the
rst termalso gives the approximation 9801
p2/4412 for , which
is correct to six decimal places. See also the more
generalRamanujanSato 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 tothe left of it equals the sum of the house numbers
to itsright. If n is between 50 and 500, what are n and x?" Thisis
a bivariate problem with multiple solutions. Ramanu-jan thought
about it and gave the answer with a twist: Hegave a continued
fraction. The unusual part was that itwas the solution to the whole
class of problems. Maha-lanobis was astounded and asked how he did
it. It issimple. The minute I heard the problem, I knew that
theanswer was a continued fraction. Which continued frac-tion, I
asked myself. Then the answer came to my mind,Ramanujan
replied.[83][84]
His intuition also led him to derive some previously un-known
identities, such as
"1 + 2
1Xn=1
cos(n)cosh(n)
#2+
"1 + 2
1Xn=1
cosh(n)cosh(n)
#2=
2434
for all , where (z) is the gamma function. Expand-ing into
series of powers and equating coecients of 0 ,4 , and 8 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 rene their formula to nd an exactconvergent series solution
to this problem. Ramanujanand Hardys work in this area gave rise to
a powerful newmethod for nding asymptotic formulae, called the
circlemethod.[85]
He discovered mock theta functions in the last year of
hislife.[86] 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: RamanujanPetersson conjecture
Although there are numerous statements that could haveborne the
name Ramanujan conjecture, there is one state-ment that was very
inuential on later work. In particu-lar, the connection of this
conjecture with conjectures ofAndrWeil 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 (q), a typical cuspform in the theory
of modular forms. It was nally provenin 1973, as a consequence of
Pierre Deligne's proof of the
-
7Weil conjectures. The reduction step involved is compli-cated.
Deligne won a Fields Medal in 1978 for his workon Weil
conjectures.[87]
4.2 Ramanujans notebooks
Further information: Ramanujans 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 nal result directly. Mathematician BruceC.
Berndt, in his review of these notebooks and Ramanu-jans 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 beeninuenced 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.[88]
The rst 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 fromRamanujans work as did G. N.Watson,B. M.
Wilson, and Bruce Berndt.[88] A fourth notebookwith 87 unorganised
pages, the so-called lost notebook,was rediscovered in 1976 by
George Andrews.[76]
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 Ramanujans birth, TIFR repub-lished the
notebooks in a coloured two-volume collectorsedition. These were
produced from scanned and mi-crolmed images of the original
manuscripts by expertarchivists of Roja Muthiah Research Library,
Chennai.
5 HardyRamanujan number 1729Main article: 1729 (number)
The number 1729 is known as the HardyRamanujannumber after a
famous anecdote of the British mathe-matician G. H. Hardy regarding
a visit to the hospital tosee Ramanujan. In Hardys words:[89]
The two dierent 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 modication ofhis hypotheses, that
were often really startling, and madehim, in his own peculiar eld,
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 Cauchys theorem,and had
indeed but the vaguest idea of what a functionof a complex variable
was....[90] 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.[91]He also stated
that he had never met his equal, and cancompare him only with Euler
or Jacobi.[91]
Quoting K. Srinivasa Rao,[92] As for his place in theworld of
Mathematics, we quote Bruce C. Berndt: 'PaulErds has passed on to
us Hardys 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 Ramanujans theo-rems
have been proven right, there had been a greater ap-preciation of
Ramanujans work and brilliance. Further,he stated Ramanujans work
was now pervading many ar-eas of modern mathematics and
physics.[86][93]
-
8 8 IN POPULAR CULTURE
In his book Scientic Edge, the physicist Jayant Narlikarspoke of
Srinivasa Ramanujan, discovered by the Cam-bridge mathematician
Hardy, whose great mathematicalndings 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 Ramanujans mathematical works.
7 Posthumous recognitionFurther information: List of things
named after SrinivasaRamanujanRamanujans home state of Tamil Nadu
celebrates 22
Bust of Ramanujan in the garden of Birla Industrial &
Techno-logical Museum.
December (Ramanujans 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 Ramanujans birth commemorating
hisachievements in the eld of number theory,[94] and a newdesign
was issued on 26 December 2011, by the IndiaPost.[95][96]
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.Ramanujans
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 inuenced 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, Ramanujans hometown, aroundRamanujans
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.[97]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 lm,chronicling the life
of Ramanujan, was released in2014 by the independent lm company
CamphorCinema.[98] The cast and crew include directorGnana
Rajasekaran, cinematographer Sunny Josephand editor B.
Lenin.[99][100] Popular Indian and En-glish stars Abhinay Vaddi,
Suhasini Maniratnam,Bhama, Kevin McGowan and Michael Lieber starin
pivotal roles.[101]
-
9 A lm, based on the book The Man Who KnewInnity: A Life of the
Genius Ramanujan byRobert Kanigel, is being made by Edward
Press-man andMatthew Brown with R. Madhavan
playingRamanujan.[102]
A feature lm is being made based on the bookThe Man Who Knew
Innity: A Life of the Ge-nius Ramanujan by Robert Kanigel; Matt
Brownwill direct on his own written script. Dev Patel willplay
Srinivasa Ramanujan, Jeremy Irons will play G.H. Hardy and Devika
Bhise will play Ramanujanswife.[103][104] Filming is set to begin
on August 3,2014 at Trinity College, Cambridge.[105]
A play, First Class Man by Alter EgoProductions,[106] was based
on David Free-mans 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 lm Tomorrow NeverDies, is working on
the lm version, starring actorSiddharth. Like the book and play it
is also titledThe First Class Man.[107]
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 ction
the events following Ramanujansletter to Hardy.[108][109]
Google honoured him on his 125th birth anniver-sary by replacing
its logo with a doodle on its homepage.[110]
Ramanujan was mentioned in the 1997 lm GoodWill Hunting, in a
scene where professor Ger-ald Lambeau (Stellan Skarsgard) explains
to SeanMaguire (Robin Williams) the genius of Will Hunt-ing (Matt
Damon) by comparing him to Ramanujan.
On 22 March 1988, the PBS Series Nova aireda documentary about
Ramanujan, The Man WhoLoved Numbers (Season 15, Episode
19).[111]
The television series Numb3rs has the character Dr.Amita
Ramanujan, a professor of applied mathe-matics, named after
Ramanujan.[112]
Ramanujans story is both referenced and echoed inCyril M.
Kornbluth's Gomez.
9 See also List of amateur mathematicians Ramanujan graph
Ramanujan summation Ramanujans constant Ramanujans ternary
quadratic form Rank of a partition 2719 (number) List of Indian
mathematicians
10 Notes[1] Berndt, Bruce C. (2005). Ramanujans Notebooks Part
V.
SpringerLink. p. 4. ISBN 0-387-94941-0.
[2] Rediscovering Ramanujan. Frontline 16 (17): 650. Au-gust
1999. Retrieved 20 December 2012.
[3] Ono, Ken (JuneJuly 2006). Honoring a Gift fromKum-bakonam
(PDF). Notices of the American MathematicalSociety (Mathematical
Association of America) 53 (6):650. Retrieved 23 June 2007.
[4] Alladi, Krishnaswami (1998). Analytic and ElementaryNumber
Theory: A Tribute to Mathematical Legend PaulErds. Norwell,
Massachusetts: Kluwer Academic Pub-lishers. p. 6. ISBN
0-7923-8273-0.
[5] Kanigel 1991, p. 11
[6] Kanigel 1991, pp. 1718
[7] Berndt & Rankin 2001, p. 89
[8] Kanigel 1991, p. 12
[9] Kanigel 1991, p. 13
[10] Kanigel 1991, p. 19
[11] Kanigel 1991, p. 14
[12] Kanigel 1991, p. 20
[13] Kanigel 1991, p. 25
[14] Berndt & Rankin 2001, p. 9
[15] 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.
[16] Kanigel 1991, p. 27
[17] Kanigel 1991, p. 39
[18] A to Z of mathematicians by Tucker McElroy 2005
ISBN0-8160-5338-3-page 221
[19] Collected papers of Srinivasa Ramanujan Srinivasa
Ra-manujan Aiyangar, Godfrey Harold Hardy, P. Vekates-vara Seshu
Aiyar 2000 ISBN 0-8218-2076-1 page xii
[20] Kanigel 1991, p. 90
[21] Kanigel 1991, p. 28
-
10 10 NOTES
[22] Kanigel 1991, p. 45
[23] Kanigel 1991, p. 47
[24] Ramanujan lost and found: a 1905 letter from TheHindu". The
Hindu (Chennai, India). 25 December 2011.
[25] Kanigel 1991, pp. 4849
[26] Kanigel 1991, pp. 5556
[27] Kanigel 1991, p. 71
[28] Ramanujans wife: Janakiammal (Janaki)" (PDF). Insti-tute of
Mathematical Sciences, Chennai. Retrieved 10November 2012.
[29] Kanigel 1991, p. 72
[30] Ramanujan, Srinivasa (1968). P. K. Srinivasan, ed.
Ra-manujan Memorial Number: Letters and Reminiscences.Madras:
Muthialpet High School. Vol. 1, p100.
[31] Kanigel 1991, p. 73
[32] Kanigel 1991, pp. 7475
[33] Ranganathan, Shiyali Ramamrita (1967). Ramanujan:The Man
and the Mathematician. Bombay: Asia Publish-ing House., p. 23.
[34] Srinivasan (1968), Vol. 1, p99.
[35] Kanigel 1991, p. 77
[36] Srinivasan (1968), Vol. 1, p129.
[37] Srinivasan (1968), Vol. 1, p86.
[38] Neville, Eric Harold (January 1921). The Late Srini-vasa
Ramanujan. Nature 106 (2673): 661662.Bibcode:1921Natur.106..661N.
doi:10.1038/106661b0.
[39] Ranganathan 1967, p. 24
[40] Kanigel 1991, p. 80
[41] Kanigel 1991, p. 86
[42] Kanigel 1991, p. 87
[43] Kanigel 1991, p. 91
[44] 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.
[45] Neville (March 1942), p292.
[46] Srinivasan (1968), p176.
[47] Srinivasan (1968), p31.
[48] Srinivasan (1968), p49.
[49] Kanigel 1991, p. 96
[50] Kanigel 1991, p. 105
[51] Letter from M. J. M. Hill to a C. L. T. Grith (a for-mer
student who sent the request to Hill on Ramanujansbehalf), 28
November 1912.
[52] Kanigel 1991, p. 106
[53] Kanigel 1991, pp. 170171
[54] Snow, C. P. (1966). Variety of Men. New York:
CharlesScribner. pp. 3031.
[55] Hardy, G. H. (June 1920). Obituary, S. Ramanujan.Nature 105
(7): 494. Bibcode:1920Natur.105..494H.doi:10.1038/105494a0.
[56] Kanigel 1991, p. 167
[57] Kanigel 1991, p. 168
[58] Hardy (June 1920), pp494495.
[59] Neville, Eric Harold (March 1942). Srini-vasa Ramanujan.
Nature 149 (3776): 293.Bibcode:1942Natur.149..292N.
doi:10.1038/149292a0.
[60] Letter, Hardy to Ramanujan, 8 February 1913.
[61] Letter, Ramanujan to Hardy, 22 January 1914.
[62] Kanigel 1991, p. 185
[63] Letter, Ramanujan to Hardy, 27 February 1913,Cambridge
University Library.
[64] Kanigel 1991, p. 175
[65] Ram, Suresh (1972). Srinivasa Ramanujan. New Delhi:National
Book Trust. p. 29.
[66] Ranganathan 1967, pp. 3031
[67] Ranganathan 1967, p. 12
[68] Kanigel 1991, p. 183
[69] Kanigel 1991, p. 184
[70] Kanigel 1991, p. 196
[71] Kanigel 1991, p. 202
[72] Hardy, G. H. (1940). Ramanujan. Cambridge:Cambridge
University Press. p. 10.
[73] Letter, Littlewood to Hardy, early March 1913.
[74] Hardy, G. H. (1979). Collected Papers of G. H.
Hardy.Oxford, England: Clarendon Press. Vol. 7, p720.
[75] Kanigel 1991, pp. 299300
[76] Peterson, Doug. Raiders of the Lost Notebook. UIUCCollege
of Liberal Arts and Sciences. Retrieved 11 Jan-uary 2014.
[77] Ramanujans Personality.
[78] Kanigel 1991, p. 36
[79] Kanigel 1991, p. 281
[80] Quote by Srinivasa Ramanujan Iyengar.
[81] Chaitin, Gregory (28 July 2007). Less Proof, MoreTruth.
NewScientist (2614): 49.
-
11
[82] Kanigel 1991, p. 283
[83] Ranganathan 1967, p. 82
[84] Calyampudi Radhakrishna Rao (1997). Statistics andtruth:
putting chance to work. World Scientic. p. 185.ISBN
978-981-02-3111-8. Retrieved 7 June 2010.
[85] Partition Formula.
[86] 100-Year-Old Deathbed Dreams of MathematicianProved True.
Fox News. 28 December 2012.
[87] Ono (JuneJuly 2006), p649.
[88] Ramanujans Notebooks.
[89] Quotations by Hardy. Gap.dcs.st-and.ac.uk. Retrieved20
November 2012.
[90] Ramanujan quote.
[91] Srinivasa Ramanujan. Retrieved 2 December 2010.
[92] K Srinivasa Rao. Srinivasa Ramanujan (22 December1887 26
April 1920)".
[93] Bruce Berndt on Ramanujans Lost Notebook, IITMadras, 24th
May 2011. youtube.com.
[94] Stamps released in 1962. Indian Postage Stamps. Re-trieved
22 May 2012.
[95] Stamps 2011. India Post. Retrieved 22 May 2012.
[96] India Post Issued a Commemorative Stamp on S Ra-manujan.
Phila Mirror. 26 December 2011. Retrieved22 May 2012.
[97] News / National :". CNN IBN. India. Retrieved 26 De-cember
2011.
[98] "'Ramanujan' Makers Shoot in His House. Indiatimes(Times
Internet Limited.). Retrieved 12 July 2013.
[99] Camphor Cinema Presents Their First FilmRamanujan.Box Oce
India. Select Publishing Company. Retrieved12 July 2013.
[100] Makers of 'Ramanujan' shoot in genius house.
http://zeenews.india.com/''. Zee Media Corporation Ltd. Re-trieved
12 July 2013.
[101] Krishnamachari, Suganthy (27 June 2013). Travails of
agenius. The Hindu (Chennai, India). Retrieved 12 July2013.
[102] Two Hollywood movies on Ramanujan soon. Sify.com.30 March
2006. Retrieved 24 July 2014.
[103] Cannes: Dev Patel to Star as Famed Indian Mathemati-cian.
hollywoodreporter.com.
[104] Jeremy Irons to Co-star in The Man Who Knew Inn-ity".
variety.com.
[105] Dev Patels The Man Who Knew Innity Moves to Pro-duction
After 8 Years in Development. variety.com.
[106] First Class Man. Alteregoproductions.org. Retrieved20
November 2012.
[107] News / National : James Bond director to make lm
onRamanujan. The Hindu (India). 16 October 2011. Re-trieved 18
October 2011.
[108] Nell Freudenberger (16 September 2007). Lust forNumbers.
The New York Times. Retrieved 4 Septem-ber 2011.
[109] DJ Taylor (26 January 2008). Adding up to a life.
TheGuardian (UK). Retrieved 4 September 2011.
[110] Google doodles for Ramanujans 125th birthday. Timesof
India. 22 December 2012. Archived from the originalon 22 December
2012. Retrieved 22 December 2012.
[111] The Man Who Loved Numbers. Pbs.org. Retrieved 18October
2011.
[112] 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 Ramanujans
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-lmed images of the
original manuscripts byexpert archivists of RojaMuthiah Research
Li-brary, Chennai.
-
12 13 EXTERNAL LINKS
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. 119146. ISBN 2-503-50673-9.
Berndt, Bruce C.; Andrews, George E. (2005).Ramanujans Lost
Notebook. Part I. New York:Springer. ISBN 0-387-25529-X.
Berndt, Bruce C.; Andrews, George E. (2008).Ramanujans Lost
Notebook. Part II. New York:Springer. ISBN 978-0-387-77765-8.
Berndt, Bruce C.; Andrews, George E. (2012).Ramanujans Lost
Notebook. Part III. New York:Springer. ISBN 978-1-4614-3809-0.
Berndt, Bruce C.; Andrews, George E. (2013).Ramanujans 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). Ramanujans Notebooks.Part I. New York:
Springer. ISBN 0-387-96110-0.
Berndt, Bruce C. (1999). Ramanujans Notebooks.Part II. New York:
Springer. ISBN 0-387-96794-X.
Berndt, Bruce C. (2004). Ramanujans Notebooks.Part III. New
York: Springer. ISBN 0-387-97503-9.
Berndt, Bruce C. (1993). Ramanujans Notebooks.Part IV. New York:
Springer. ISBN 0-387-94109-6.
Berndt, Bruce C. (2005). Ramanujans 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-nity: a Life of the
Genius Ramanujan. New York:Charles Scribners 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):
15191521.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). Scientic 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 Ramanujans 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 (18871920) from
ScienceWorld.
-
13.3 Other links 13
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.
Hindu.com: Norwegian and Indian mathematicalgeniuses, RAMANUJAN
Essays and Surveys,Ramanujans growing inuence, Ramanujans
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):
595601.doi:10.2307/2589114. JSTOR 2589114. MR1786233.
Ramanujans mock theta function puzzle solved Ramanujans 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 licenses14.1
Text
Srinivasa Ramanujan Source:
http://en.wikipedia.org/wiki/Srinivasa_Ramanujan?oldid=663800698
Contributors: AxelBoldt, TheAnome, Jeronimo, XJaM, William Avery,
AdamRetchless, Pichai Asokan, Michael Hardy, Shyamal, Gabbe,
Menchi, Chinju, Lquilter,GTBacchus, Mcarling, Paddu, Card~enwiki,
Ahoerstemeier, Ijon, Sray, Poor Yorick, Jengod, JidGom, Charles
Matthews, NilsB, Dfeuer,Nataraja~enwiki, Jitse Niesen, Zoicon5,
Tpbradbury, Ldrhcp, Furrykef, Nv8200pa, Taxman, VeryVerily, Phys,
Topbanana, AnonMoos,Eugene van der Pijll, Chuunen Baka, Robbot,
Vincent Gray, Sander123, Jaredwf, Fredrik, Chocolateboy, Lowellian,
DHN, Jondel, An-drew Levine, Rrjanbiah, JackofOz, Ambarish,
Anthony, Mattaschen, MikeCapone, Ancheta Wis, Exploding Boy,
Giftlite, Nichalp, Vish-vas vasuki, var Arnfjr Bjarmason, Lethe,
Berasategui, Peruvianllama, Everyking, Gro-Tsen, Curps, Gamaliel,
Jason Quinn, Neilc,Etaonish, Zarvok, Iceager, DragonySixtyseven,
Thincat, Jawed, Icairns, Urhixidur, Ukexpat, Robin klein, Karl
Dickman, Abdull, Ja-son Carreiro, Shahab, Venu62, Indosauros,
Pyrop, RossPatterson, Discospinster, Rich Farmbrough, Guanabot,
FT2, Vsmith, ArthurHolland, Ashwatham, Paul August, Bender235,
Billlion, CanisRufus, Alren, Pt, VishalB, Szquirrel, RoyBoy,
AlvinMGO, Gershwinrb,Causa sui, Bobo192, Evolauxia, Blakkandekka,
Arcadian, I9Q79oL78KiL0QTFHgyc, La goutte de pluie, ,
Photonique,Larry V, LostLeviathan, Helix84, Haham hanuka,
JesseHogan, Merope, Ranveig, Jumbuck, Alansohn, Arthena, Salilb,
Riana, SlimVir-gin, Gaurav1146, Mrholybrain, NTK, LavosBacons,
Wtmitchell, Velella, Hasdrubal~enwiki, VivaEmilyDavies, Staeiou,
Vadakkan, Ra-jprem, Oleg Alexandrov, Zntrip, Velho, Jak86,
Woohookitty, Linas, Mindmatrix, Shreevatsa, Malapati, Pol098,
Dodiad, Julien Tuer-linckx, Mpatel, Slocombe, Code Wizard,
Nandakumarg, Markizs~enwiki, 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,
DonSiano, R.e.b., Sango123, Aveekbh, Fish and karate, Baryonic
Being, Mtommila, DaStressor, Maxal, Pete.Hurd, Alphachimp,
Themissinglint, Glenn L, Gareth E Kegg, IntrepidWill, Haonhien,
Chobot, DaGizza, Jagdeep,Adoniscik, EamonnPKeane, YurikBot,
Wavelength, RobotE, Ventolin, Conscious, Zaroblue05, Lenthe,
Thoreaulylazy, Gaius Cornelius,Member, NawlinWiki, Rak3sh,
Shreshth91, Thatdog, Lowe4091, Chick Bowen, Stompbox, Banes, DYLAN
LENNON~enwiki, MoeEpsilon, Mikeblas, Hv, Jalabi99, Nick C, Tony1,
Zwobot, Supten, Samir, Gadget850, DeadEyeArrow, Bota47, Haemo,
Black Falcon,Gnusbiz, Wknight94, Ms2ger, Crisco 1492, Zargulon,
Ario, Paul Magnussen, Ninly, Bhumiya, Djkimmons, Arthur Rubin,
Sarefo, Tevildo,LeonardoRob0t, Shyam, Arundhati bakshi, Pred,
Garion96, Ilmari Karonen, Chitrada, Katieh5584, Kungfuadam, Banus,
Cyphase, DVDRW, Vineethtm, Sintonak.X, GrafZahl, SmackBot,
Drummondjacob, YellowMonkey, Roger Hui, Lestrade, Herostratus,
InverseHypercube,Pgk, Jagged 85, Thunderboltz, Eskimbot, Dexter73,
Alsandro, SmartGuy Old, Msrkiran, Commander Keane bot, Eiler7,
Xaosux, Gilliam,Andy M. Wang, Rst20xx, Bvssatish, Pseinstein,
Vignesh.ks, Zouf, Bluebot, Droll, Acid2base, TheFeds, Chisophugis,
Exitr, Nbarth, Whis-pering, Colonies Chris, Lenin and McCarthy,
OrphanBot, Onorem, Vanished User 0001, Gala.martin, LouScheer,
Cribananda, Hyper-geometric2F1(a,b,c,x), Robma, Fuhghettaboutit,
Jackohare, DRLB, Allansteel, G716, Kshieh, Zzorse, N Shar,
Pilotguy, Vaibhavgarg,Ohconfucius, RNLion, Lambiam, Nishkid64,
ArglebargleIV, Mksword, Ser Amantio di Nicolao, Gloriamarie,
Harryboyles, Mouse Night-shirt, Saravan p, Rohit math, Titus III,
Richard L. Peterson, Wtwilson3, Disavian, Sir Nicholas de
Mimsy-Porpington, Bucksburg, Shyam-sunder, Mgiganteus1, Syrcatbot,
SamuelLurie, Ashokmani~enwiki, BillFlis, Yvesnimmo, Kseferovic,
Mets501, Groovybill, Sijo Ripa,Dr.K., Rlinnity, Xionbox,
ShakingSpirit, KJS77, Hu12, Norm mit, Balu.muthu, Clarityend, Paul
venter, JoeBot, Onathinwhiteline,Diddyeinstein, Tawkerbot2, Bubbha,
Ioannes Pragensis, Thedemonhog, Randhirreddy, Dycedarg, Van
helsing, MoxJet, DSachan, Alex-conlin, Nczempin, Gyopi, THF,
AshLin, Outriggr, Andkore, Chicheley, Myasuda, Stormwyrm, Cydebot,
Resham Sivnarain, Gogo Dodo,Karlegerton, Ibroy00, Peter439, Mon4,
ST47, M a s, DumbBOT, Omexis, Thijs!bot, Epbr123, 271828182,
Andyjsmith, Mungomba,Nonagonal Spider, Headbomb, James086,
Dankster, Escarbot, LachlanA, Hmrox, AntiVandalBot, The Obento
Musubi, RobotG, Wain-son, Yonatan, P.K.Niyogi, Why My Fleece?,
CPWinter, Opelio, Autocracy, Julia Rossi, Exteray, Coyets, NSH001,
Hannes Eder, Myanw,MikeLynch, Paul1776, Husond, Bakasuprman, MER-C,
Epeeeche, Arch dude, Albany NY, 100110100, Chadhade, AtticusX,
Soul-bot, Ian.homer~enwiki, Tuncrypt, Spellmaster, Parunach, Kope,
DerHexer, JaGa, WLU, DancingPenguin, Bsvprasad, MartinBot,
Vino-grad19, Ravichandar84, Glrx, CommonsDelinker, Fconaway,
DBlomgren, MacAuslan, Pomte, Jargon777, Gotyear, Thadriel,
Whaatt,SuperGirl, Ttwo, Maurice Carbonaro, Interplanet Janet,
Gowthamjayapal, Aviad2001, Hair Commodore, Salih, Bigmac31,
Ignatzmice, Kr-ishnachandranvn, In Transit, Robertgreer, Sigmundur,
Joshua Issac, DavidCBryant, Donmike10, Michael Angelkovich, Yecril,
Redtigerxyz,VolkovBot, Taxiarchos228, ToyotaPanasonic, Je G.,
AlnoktaBOT, Satani, Gpohara, Petello12000, Philip Trueman,
TXiKiBoT, Rajiv-sundar, Vipinhari, Technopat, Rei-bot, Anonymous
Dissident, Pandacomics, Macslacker, Voorlandt, Gauge00, DragonLord,
Praveen pillay,Ripepette, Primes, David in DC, BigDunc, Enigmaman,
VanishedUserABC, Romanator14, TheDevils Advocate, Naa.ganesan,
Moral Sim-plex, Logan, Resurgent insurgent, EmxBot, Subh83, Dogah,
SieBot, StAnselm, Rajamankkan, Madman, TJRC, Badripk, Caltas,
Yintan,JabbaTheBot, Thai H. Nguyen, Adiswini, Renatops, Arda Xi,
Bentogoa, Oxymoron83, Antonio Lopez, Avnjay, R0uge, Polbot,
FranciscoTevez, AlanUS, RogueTeddy, Aumnamahashiva, Alwpoe,
AtomikWeasel, Virdisukhvir, Randomblue, Hindduking, Sitush,
TheCatalyst31,Ambi saba, Atif.t2, Martarius, ClueBot, Justin W
Smith, Plastikspork, RODERICKMOLASAR, Nazra, TypoBoy, WestwoodMatt,
Mas-terpiece2000, Excirial, Nymf, Relata refero, Csesznekgirl,
Teju2friends, Estirabot, Sun Creator, Wprlh, Razorame, Jsondow, Ark
pillai,MelonBot, SoxBot III, Indopug, NERIC-Security, Yun-Yuuzhan
(lost password), QwertySama, Mbhagavan, Kaustubh, Jovianeye,
Zo-poric, WikHead, Tmsankaran, Asdondia, Addbot, Kvishwanathr, DOI
bot, Opus88888, Thaejas, Ronhjones, Trippt, Goalieprice13,Libanar,
Favonian, LinkFA-Bot, Squandermania, 84user, Lightbot, Xenobot,
Webwizard, Mohammed sultan, Legobot, Mik01aj, Luckas-bot, Yobot,
Jason Recliner, Esq., Kseon, Arkachatterjea, K2709, AnomieBOT,
Fatal!ty, Jim1138, Tucoxn, Keithbob, Materialscien-tist, uman,
Citation bot, ArthurBot, Krishnakoli, BalajiRamasubramanian, Xqbot,
Jayarathina, TinucherianBot II, Gilo1969, Gap9551,Srich32977,
Ariaravelan, Ruy Pugliesi, Frosted14, Ash deb, Omnipaedista,
Raulshc, Skonhet1976, Misortie, Deikiw, FrescoBot, Nageh,MilanJain,
LavenhamHistorian, Mathprofrockstar, Grandiose, John85, Atlantia,
Citation bot 1, MarB4, Aldy, SpacemanSpi, I dreamof horses,
Indythegeek, MarcelB612, SpaceFlight89, Sinsen, Prabaraj,
Foobarnix, Plasticspork, Barras, Kgrad, FoxBot, TobeBot,
Pra-matheshb, LogAntiLog, Gulbenk, Oracleofottawa, Defender of
torch, Kobeisbeast, An Justied Wikipedian, Hari7478, Tbhotch,
DARTHSIDIOUS 2, Onel5969, RjwilmsiBot, TjBot, Timepassharsha,
DRAGON BOOSTER, WildBot, Salvio giuliano, Djmdutch11, Hunny-narain,
EmausBot, John of Reading, WikitanvirBot, Super48paul, Drankush,
Vanished user bc8e8hwkjahw8tijwu, LRP;840, Wikipelli,Incrazy,
Anirudh Emani, Kkm010, ZroBot, Cupidvogel, Vensatry, Subhom.mitra,
Elite ferns, Flies 1, AOC25, H3llBot, Suslindisam-biguator, Tech
Cello, Kyucasio, QEDK, Keyan20, Karthikndr, Brandmeister,
YnnusOiramo, Narayannat, Mayur, Donner60, Avmpsy-cho, Pun,
ChuispastonBot, Iamjamesbond007, Firstblossoms, Llightex,
Rmashhadi, DemonicPartyHat, 28bot, Maxkingesq,
Vijay184,Sonicyouth86, Slimepot636, Gwen-chan, Kunalkishore55,
ClueBot NG, Aslamwonders, Gareth Grith-Jones, El Roih,
MelbourneStar,LogX, Satellizer, Tgrmurthy, LJosil, Tgramamurthy,
O.Koslowski, Rajeshmondal18, Gganesheng, Kasirbot, Widr,
Er.avijit.pandey, Euty,Sriharanathrayala, .GeeSharma, Vibhijain,
Helpful Pixie Bot, Thisthat2011, Titodutta, Bibcode Bot, DBigXray,
BG19bot, Arisimandi,Isengupta, MusikAnimal, Alangar Manickam,
Akhilesh92, Mark Arsten, Ninney, StickyWikis, Benzband,
Giorgos.antoniou, Rajeev415,Hawker07, MythosMagic,
Ramprakashpalaparty, Robert Goddard, MThekkumthala, Vanischenu,
Marcokrt, BattyBot, Sumitkumarjha75,
-
14.2 Images 15
Justincheng12345-bot, Pratyya Ghosh, Arr4, Cyberbot II,
Ancienzus, Cpt.a.haddock, Soni, Dexbot, Jaspermogg, Magentic
Manifesta-tions, Webclient101, Nandagopal.v, Aumkaar Pranav,
Aloak1, Siberian Patriot, Nikhilvsrockz, Frosty, Graphium,
HullIntegrity, DannySprinkle, Ranjyotiprakashsingh, Dravidianhero,
PinkAmpersand, TamBram, Epicgenius, Missionedit, Piesiva,
Irahulpandey, Abhiraj m,Jamesmcmahon0, Eshwar.om, Sosthenes12,
Jakec, EvergreenFir, Gouthams2, Timpu454, Whiteuyduck, Babitaarora,
Rajashree Ra-jaram, The Herald, AioftheStorm, Ginsuloft,
Pratik.patra, Kartik61182, Varun.a8, Aryan203, ShivamPatel
scientist, Dierentthought-tomaths, Monkbot, Akpelirw, Skwwtlu,
Ssven2, Alex20141729, Ice ax1940ice pick, Karyakarta, Sanjeev2015,
TanmayDaga, Vijay-jeyasanksr, Haywardcir, SSoheilHosseini,
D.v.karthick, Rahulgurujala, Vivek Sarje, GangGangQuokka2, Aniroodh
Sarkar, Thelimiter,Suresh151622, Daivaghna, BarnQuest, Supdiop,
Samanvaylamba, Agrawalsanskar, Prateek Proo, Rishabh23599 and
Anonymous: 985
14.2 Images File:Commons-logo.svg Source:
http://upload.wikimedia.org/wikipedia/en/4/4a/Commons-logo.svg
License: ? Contributors: ? Original
artist: ? 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
File:Speakerlink-new.svg Source:
http://upload.wikimedia.org/wikipedia/commons/3/3b/Speakerlink-new.svg
License: CC0 Contribu-tors: Own work Original artist:
Kelvinsong
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
File:Srinivasa_Ramanujan_-_OPC_-_1.jpg Source:
http://upload.wikimedia.org/wikipedia/commons/c/c1/Srinivasa_Ramanujan_-_OPC_-_1.jpg
License: CC BY-SA 2.0 de Contributors: Oberwolfach Photo
Collection, original location Original artist: Konrad Jacobs
File:Srinivasa_Ramanujan_signature.gif Source:
http://upload.wikimedia.org/wikipedia/commons/1/18/Srinivasa_Ramanujan_signature.gif
License: Public domain Contributors:
http://www.imsc.res.in/~{}rao/ramanujan/introindex.html Original
artist: SrinivasaRamanujan
File:Whewell{}s_Court,_Trinity_College,_Cambridge.jpg Source:
http://upload.wikimedia.org/wikipedia/commons/7/77/Whewell%27s_Court%2C_Trinity_College%2C_Cambridge.jpg
License: CC BY-SA 2.0 Contributors: From geograph.org.uk
Originalartist: Stanley Howe
File:Wikiquote-logo.svg Source:
http://upload.wikimedia.org/wikipedia/commons/f/fa/Wikiquote-logo.svg
License: Public domainContributors: ? Original artist: ?
File:Wikisource-logo.svg Source:
http://upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg
License: CC BY-SA 3.0Contributors: Rei-artur Original artist:
Nicholas Moreau
14.3 Content license Creative Commons Attribution-Share Alike
3.0
Early lifeAdulthood in IndiaAttention towards
mathematicsContacting English mathematicians
Life in EnglandIllness and return to IndiaPersonality and
spiritual life
Mathematical achievementsThe Ramanujan conjectureRamanujans
notebooks
HardyRamanujan number 1729Other mathematicians views of
RamanujanPosthumous recognitionIn popular cultureSee
alsoNotesSelected publications by RamanujanSelected publications
about Ramanujan and his workExternal linksMedia linksBiographical
linksOther links
Text and image sources, contributors, and
licensesTextImagesContent license
