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{{Short description|Indian mathematician (1887–1920)}}
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{{Indian name|Ramanujan|Srinivasa}}
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{{Use British English|date=September 2014}}
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{{Infobox scientist
| caption           = Ramanujan in 1913
| name              = Srinivasa Ramanujan
| honorific_suffix  = {{postnominals|FRS|size=100}}
| image             = File:Srinivasa Ramanujan-Add. MS a94 version2 (cropped).jpg
| birth_name        = Srinivasa Ramanujan Aiyangar
| birth_date        = {{Birth date|df=yes|1887|12|22}}
| birth_place       = [[Erode]], [[Mysore Princely State|Mysore State]], [[British Raj|British India]] (now in [[Tamil Nadu]], India)
| death_date        = {{Death date and age|df=yes|1920|4|26|1887|12|22}}
| death_place       = [[Kumbakonam]], [[Tanjore District (Madras Presidency)|Tanjore District]], [[Madras Presidency]], [[British Raj|British India]] (now [[Thanjavur district]],<br />[[Tamil Nadu]], India)
| citizenship       = [[British Raj|British Indian]]
| workplaces        = [[University of Cambridge]]
| education         = {{plainlist|
* [[Government Arts College, Kumbakonam|Government Arts College]]
* [[Pachaiyappa's College]]
* {{nowrap|[[Trinity College, Cambridge]] {{nowrap|([[Bachelor of Arts|BA]])}}}}}}
| thesis_title      = Highly Composite Numbers
| thesis_url        = https://web.archive.org/web/20180809110104/http://math.univ-lyon1.fr/~nicolas/ramanujanNR.pdf
| thesis_year       = 1916
| academic_advisors = {{plainlist|
* [[G. H. Hardy]]
* [[John Edensor Littlewood|J. E. Littlewood]]}}
| known_for         = {{plainlist|
* [[Ramanujan's sum]]
* [[Landau–Ramanujan constant]]
* [[Mock theta function]]s
* [[Ramanujan's congruences]]
* [[Ramanujan conjecture]]
* [[Ramanujan prime]]
* [[Ramanujan–Soldner constant]]
* [[Ramanujan theta function]]
* [[Rogers–Ramanujan identities]]
* [[Ramanujan's master theorem]]
* [[Partition function (number theory)#Approximation formulas|Hardy–Ramanujan asymptotic formula]]
* [[Ramanujan–Sato series]]}}
| awards            = [[Fellow of the Royal Society]] (1918)
| signature         = Signature of Srinivasa Ramanujan.svg
| signature_alt     = Srinivasa Ramanujan signature
| field             = [[Mathematics]]
| doctoral_advisors = <!--the PhD did not exist at Cambridge until 1919-->
}}
'''Srinivasa Ramanujan Aiyangar'''{{efn|{{postnominals |FRS}} ({{IPAc-en|ˈ|s|r|iː|n|ᵻ|v|ɑː|s|ə|_|r|ɑː|ˈ|m|ɑː|n|ʊ|dʒ|ən}} {{respell|SREE|nih|vah|sə|_|rah|MAH|nuuj|ən}};<ref>{{cite book|last1= Olausson|first1=Lena|last2=Sangster|first2= Catherine|year=2006|title=Oxford BBC Guide to Pronunciation|publisher= Oxford University Press|page=322|isbn= 978-0-19-280710-6}}</ref> born '''Srinivasa Ramanujan Aiyangar''', {{IPA|ta|sriːniʋaːsa ɾaːmaːnud͡ʑan ajːaŋgar|lang}})<ref name="ODNB-Ramanujan">{{cite ODNB|id=51582|title= Ramanujan, Srinivasa |year=2004 |first=Robert|last= Kanigel}}</ref><ref>{{Cite web|url=https://trove.nla.gov.au/people/895585?c=people|title=Ramanujan Aiyangar, Srinivasa (1887–1920)
|website=trove.nla.gov.au}}</ref>}}
(22 December 1887{{spaced ndash}}26 April 1920) was an Indian [[mathematician]]. Often regarded as one of the greatest mathematicians of all time, though he had almost no formal training in [[pure mathematics]], he made substantial contributions to [[mathematical analysis]], [[number theory]], [[infinite series]], and [[continued fraction]]s, including solutions to mathematical problems then considered unsolvable.

Ramanujan initially developed his own mathematical research in isolation. According to [[Hans Eysenck]], "he tried to interest the leading professional mathematicians in his work, but failed for the most part. What he had to show them was too novel, too unfamiliar, and additionally presented in unusual ways; they could not be bothered".<ref>[[Hans Eysenck]] (1995). ''Genius'', p. 197. Cambridge University Press, {{ISBN|0-521-48508-8}}.</ref> Seeking mathematicians who could better understand his work, in 1913 he began a mail correspondence with the English mathematician [[G. H. Hardy]] at the [[University of Cambridge]], England. Recognising Ramanujan's work as extraordinary, Hardy arranged for him to travel to Cambridge. In his notes, Hardy commented that Ramanujan had produced groundbreaking new [[theorem]]s, including some that "defeated me completely; I had never seen anything in the least like them before",<ref>{{cite book |last1=Hardy |first1=Godfrey Harold |year=1940 |title=Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work |publisher=Cambridge University Press |page=9 |isbn=0-8218-2023-0}}</ref> and some recently proven but highly advanced results.

During his short life, Ramanujan independently compiled nearly 3,900 results (mostly [[identity (mathematics)|identities]] and [[equation]]s).<ref>{{Cite book |title= Ramanujan's Notebooks |volume = Part 5 |last= Berndt |first=Bruce C. |date=12 December 1997 |publisher= Springer Science & Business |isbn=978-0-38794941-3 |page = 4}}</ref> Many were completely novel; his original and highly unconventional results, such as the [[Ramanujan prime]], the [[Ramanujan theta function]], [[partition (number theory)|partition]] formulae and [[mock theta function]]s, have opened entire new areas of work and inspired further research.<ref name=ono>{{cite journal |author-link = Ken Ono |last=Ono |first=Ken |date= June–July 2006 |title= Honoring a Gift from Kumbakonam |journal= [[Notices of the American Mathematical Society]] |volume= 53 |issue= 6 |pages= 640–51 [649–50] |url= https://www.ams.org/notices/200606/fea-ono.pdf |access-date=2007-06-23 |url-status= live |archive-url=https://web.archive.org/web/20070621074858/https://www.ams.org/notices/200606/fea-ono.pdf |archive-date= 21 June 2007}}</ref> Of his thousands of results, most have been proven correct.<ref name="frontline">{{cite journal |date= August 1999 |title= Rediscovering Ramanujan |journal= [[Frontline (magazine)|Frontline]] |volume= 16 |issue= 17 |page= 650 |url= https://www.frontline.in/static/html/fl1617/16170810.htm |access-date= 20 December 2012 |url-status= dead |archive-url= https://web.archive.org/web/20130925201456/http://www.frontline.in/static/html/fl1617/16170810.htm |archive-date= 25 September 2013}}</ref> ''[[The Ramanujan Journal]]'', a [[scientific journal]], was established to publish work in all areas of mathematics influenced by Ramanujan,<ref>{{cite book |title=Analytic and Elementary Number Theory: A Tribute to Mathematical Legend Paul Erdos |last1 = Alladi |first1 = Krishnaswami |last2 = Elliott |first2 = P. D. T. A.| last3 = Granville |first3 = A. |date=30 September 1998 |publisher=Springer Science & Business |isbn= 978-0-79238273-7 |page = 6}}</ref> and his notebooks—containing summaries of his published and unpublished results—have been analysed and studied for decades since his death as a source of new mathematical ideas. As late as 2012, researchers continued to discover that mere comments in his writings about "simple properties" and "similar outputs" for certain findings were themselves profound and subtle number theory results that remained unsuspected until nearly a century after his death.<ref name= "ono_simpleproperties">{{Cite web|url=https://www.newscientist.com/article/dn20039-deep-meaning-in-ramanujans-simple-pattern/|archiveurl=https://web.archive.org/web/20170803140127/https://www.newscientist.com/article/dn20039-deep-meaning-in-ramanujans-simple-pattern/|url-status=dead|title=Deep meaning in Ramanujan's 'simple' pattern|archivedate=3 August 2017|website=New Scientist}}</ref><ref name="ono_mock">[https://www.newscientist.com/article/mg21628904-200-mathematical-proof-reveals-magic-of-ramanujans-genius/ "Mathematical proof reveals magic of Ramanujan's genius"] {{webarchive |url= https://web.archive.org/web/20170709084351/https://www.newscientist.com/article/mg21628904-200-mathematical-proof-reveals-magic-of-ramanujans-genius/|date=9 July 2017}}. ''[[New Scientist]]''.</ref> He became one of the youngest [[Fellow of the Royal Society|Fellows of the Royal Society]] and only the second Indian member, and the first Indian to be elected a [[Trinity College, Cambridge#Notable fellows and alumni|Fellow of Trinity College, Cambridge]].

In 1919, ill health—now believed to have been hepatic [[amoebiasis]] (a complication from episodes of [[dysentery]] many years previously)—compelled Ramanujan's return to India, where he died in 1920 at the age of 32. His last letters to Hardy, written in January 1920, show that he was still continuing to produce new mathematical ideas and theorems. His "[[Ramanujan's lost notebook|lost notebook]]", containing discoveries from the last year of his life, caused great excitement among mathematicians when it was rediscovered in 1976.

==Early life==
[[File:Erode, 18 Alahiri Str - Ramanujan birth place.jpg|thumb|upright|Ramanujan's birthplace on 18 Alahiri Street, [[Erode]], now in [[Tamil Nadu]]]]
[[File:Ramanujanhome.jpg|thumb|upright|Ramanujan's home on Sarangapani Sannidhi Street, [[Kumbakonam]]]]
Ramanujan (literally, "younger brother of [[Rama]]", a Hindu deity)<ref name="Kanigel-1991-12">{{harvnb|Kanigel|1991|page=12}}</ref> was born on 22 December 1887 into a [[Tamil Brahmin]] [[Iyengar]] family in [[Erode]], in present-day [[Tamil Nadu]].<ref>{{harvnb|Kanigel|1991|page=11}}</ref> His father, Kuppuswamy Srinivasa Iyengar, originally from [[Thanjavur district]], worked as a clerk in a [[sari]] shop.<ref>{{harvnb|Kanigel|1991|pages=17–18}}</ref><ref name="ODNB-Ramanujan" /> His mother, Komalatammal, was a [[housewife]] and sang at a local temple.<ref>{{Harvnb|Berndt|Rankin|2001|p=89}}</ref> They lived in a small traditional home on Sarangapani Sannidhi Street in the town of [[Kumbakonam]].<ref name="hindu">{{cite news|last1=Srinivasan|first1=Pankaja|title=The Nostalgia Formula|url=https://www.thehindu.com/features/friday-review/history-and-culture/the-nostalgia-formula/article4013282.ece|access-date=7 September 2016|newspaper=The Hindu|date=19 October 2012}}</ref> The family home is now a museum. When Ramanujan was a year and a half old, his mother gave birth to a son, Sadagopan, who died less than three months later. In December 1889, Ramanujan contracted [[smallpox]], but recovered, unlike the 4,000 others who died in a bad year in the Thanjavur district around this time. He moved with his mother to her parents' house in [[Kanchipuram]], near Madras (now [[Chennai]]). His mother gave birth to two more children, in 1891 and 1894, both of whom died before their first birthdays.<ref name="Kanigel-1991-12" />

On 1 October 1892, Ramanujan was enrolled at the local school.<ref>{{harvnb|Kanigel|1991|page=13}}</ref> After his maternal grandfather lost his job as a court official in Kanchipuram,<ref>{{harvnb|Kanigel|1991|page=19}}</ref> Ramanujan and his mother moved back to [[Kumbakonam]], and he was enrolled in Kangayan Primary School.<ref name="Kanigel-1991-14">{{harvnb|Kanigel|1991|page=14}}</ref> When his paternal grandfather died, he was sent back to his maternal grandparents, then living in Madras. He did not like school in Madras, and tried to avoid attending. His family enlisted a local constable to make sure he attended school. Within six months, Ramanujan was back in Kumbakonam.<ref name="Kanigel-1991-14" />

Since Ramanujan's father was at work most of the day, his mother took care of the boy, and they had a close relationship. From her, he learned about tradition and [[puranas]], to sing religious songs, to attend [[Puja (Hinduism)|pujas]] at the temple, and to maintain particular eating habits—all part of [[Brahmin]] culture.<ref>{{harvnb|Kanigel|1991|page=20}}</ref> At Kangayan Primary School, Ramanujan performed well. Just before turning 10, in November 1897, he passed his primary examinations in English, [[Tamil language|Tamil]], geography, and arithmetic with the best scores in the district.<ref name="Kanigel-1991-25">{{harvnb|Kanigel|1991|page=25}}</ref> That year, Ramanujan entered [[Town Higher Secondary School]], where he encountered formal mathematics for the first time.<ref name="Kanigel-1991-25" />

A [[child prodigy]] by age 11, he had exhausted the mathematical knowledge of two college students who were lodgers at his home. He was later lent a book written by [[S. L. Loney]] on advanced trigonometry.<ref name="berndt9">{{Harvnb|Berndt|Rankin|2001|p=9}}</ref><ref>{{Cite book| last=Hardy | first=G. H.| title=Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work| publisher=[[American Mathematical Society]] | location=Providence, Rhode Island | year = 1999|isbn=978-0-8218-2023-0 | page=2 }}</ref> He mastered this by the age of 13 while discovering sophisticated theorems on his own. By 14, he received merit certificates and academic awards that continued throughout his school career, and he assisted the school in the logistics of assigning its 1,200 students (each with differing needs) to its approximately 35 teachers.<ref name="Kanigel-1991-27">{{harvnb|Kanigel|1991|page=27}}</ref> He completed mathematical exams in half the allotted time, and showed a familiarity with [[geometry]] and [[infinite series]]. Ramanujan was shown how to solve cubic equations in 1902. He would later develop his own method to solve the [[Quartic function|quartic]]. In 1903, he tried to solve the [[Quintic function|quintic]], not knowing that it was [[Abel–Ruffini theorem|impossible to solve with radicals.]]<ref>{{Cite web |title=Srinivasa Ramanujan - Biography |url=https://mathshistory.st-andrews.ac.uk/Biographies/Ramanujan/ |access-date=2022-10-29 |website=Maths History |language=en}}</ref>

In 1903, when he was 16, Ramanujan obtained from a friend a library copy of ''[[Synopsis of Pure Mathematics|A Synopsis of Elementary Results in Pure and Applied Mathematics]]'', [[G. S. Carr]]'s collection of 5,000 theorems.<ref>{{harvnb|Kanigel|1991|page=39}}</ref><ref>McElroy, Tucker (2005). ''A to Z of mathematicians''. Facts on File. p. 221. {{isbn|0-8160-5338-3-}}</ref> Ramanujan reportedly studied the contents of the book in detail.<ref name=papers>{{citation | title= Collected papers of Srinivasa Ramanujan | journal=Nature | volume=123 | issue=3104 | first1=Srinivasa | last1=Ramanujan Aiyangar | first2= Godfrey Harold | last2= Hardy | first3= P. Veṅkatesvara Seshu |last3=Aiyar |year= 2000 |isbn= 978-0-8218-2076-6 |page= xii| bibcode=1929Natur.123..631L | doi=10.1038/123631a0 | s2cid=44812911 }}</ref> The next year, Ramanujan independently developed and investigated the [[Bernoulli number]]s and calculated the [[Euler–Mascheroni constant]] up to 15 decimal places.<ref>{{harvnb|Kanigel|1991|page=90}}</ref> His peers at the time said they "rarely understood him" and "stood in respectful awe" of him.<ref name="Kanigel-1991-27" />

When he graduated from Town Higher Secondary School in 1904, Ramanujan was awarded the K. Ranganatha Rao prize for mathematics by the school's headmaster, Krishnaswami Iyer. Iyer introduced Ramanujan as an outstanding student who deserved scores higher than the maximum.<ref name="Kanigel-1991-unknown-page">{{harvnb|Kanigel|1991|page=??}}</ref> He received a scholarship to study at [[Government Arts College, Kumbakonam]],<ref name="Kanigel-1991-28">{{harvnb|Kanigel|1991|page=28}}</ref><ref name="Kanigel-1991-45">{{harvnb|Kanigel|1991|page=45}}</ref> but was so intent on mathematics that he could not focus on any other subjects and failed most of them, losing his scholarship in the process.<ref name="Kanigel-1991-47-48">{{harvnb|Kanigel|1991|pages=47–48}}</ref> In August 1905, Ramanujan ran away from home, heading towards [[Visakhapatnam]], and stayed in [[Rajahmundry]]<ref>{{cite news | url=https://www.thehindu.com/features/friday-review/history-and-culture/ramanujan-lost-and-found-a-1905-letter-from-the-hindu/article2745164.ece | location=Chennai, India | work=The Hindu | title=Ramanujan lost and found: a 1905 letter from ''The Hindu'' | date=25 December 2011 }}</ref> for about a month.<ref name="Kanigel-1991-47-48" /> He later enrolled at [[Pachaiyappa's College]] in Madras. There, he passed in mathematics, choosing only to attempt questions that appealed to him and leaving the rest unanswered, but performed poorly in other subjects, such as English, physiology, and Sanskrit.<ref name="hindu2">{{cite news|last1=Krishnamachari|first1=Suganthi|title=Travails of a Genius|url=https://www.thehindu.com/features/friday-review/travails-of-a-genius/article4857108.ece|access-date=7 September 2016|newspaper=The Hindu|date=27 June 2013|url-status=live|archive-url=https://web.archive.org/web/20170826080524/https://www.thehindu.com/features/friday-review/travails-of-a-genius/article4857108.ece|archive-date=26 August 2017}}</ref> Ramanujan failed his [[Royal Society of Arts|Fellow of Arts]] exam in December 1906 and again a year later. Without an FA degree, he left college and continued to pursue independent research in mathematics, living in extreme poverty and often on the brink of starvation.<ref name="Kanigel-1991-55-56">{{harvnb|Kanigel|1991|page=55–56}}</ref>

In 1910, after a meeting between the 23-year-old Ramanujan and the founder of the [[Indian Mathematical Society]], [[V. Ramaswamy Aiyer]], Ramanujan began to get recognition in Madras's mathematical circles, leading to his inclusion as a researcher at the [[University of Madras]].<ref name="krish">{{cite web|last1=Krishnamurthy|first1=V.|title=Srinivasa Ramanujan – His life and his genius|url=https://www.krishnamurthys.com/profvk/VK2/Srinivasa_Ramanujan.html|website=www.krishnamurthys.com|publisher=(Expository address delivered on Sep.16, 1987 at Visvesvarayya Auditorium as part of the celebrations of Ramanujan Centenary by the IISC, Bangalore)|access-date=7 September 2016|url-status=dead|archive-url=https://web.archive.org/web/20160921182431/https://www.krishnamurthys.com/profvk/VK2/Srinivasa_Ramanujan.html|archive-date=21 September 2016}}</ref>

==Adulthood in India==
On 14 July 1909, Ramanujan married Janaki (Janakiammal; 21 March 1899 – 13 April 1994),<ref>{{cite web | work = Live mint |url= https://www.livemint.com/Leisure/6vJOM00P1sGZgCxV7RIasI/The-seamstress-and-the-mathematician.html|title=The seamstress and the mathematician|date= 20 April 2018}}</ref> a girl his mother had selected for him a year earlier and who was ten years old when they married.<ref name="Kanigel-1991-71">>{{harvnb|Kanigel|1991|page=71}}</ref><ref name= "History in adult human sexual behavior with children and adolescents in Western societies">{{cite book |last1=Bullough|first1= V.L. | chapter = 2. History in adult human sexual behavior with children and adolescents in Western societies|date= 1990 |publisher= Springer-Verlag | place = New York |isbn= 978-1-46139684-0 |pages=71| title = Pedophilia: Biosocial Dimensions|chapter-url= https://books.google.com/books?id=nnKEBwAAQBAJ&pg=PA69}}</ref><ref>{{Cite journal |last= Kolata |first=Gina |author-link= Gina Kolata |date=19 June 1987 |title= Remembering a 'Magical Genius' |journal= Science |series= New Series |volume=236 |issue=4808 |pages= 1519–21 |doi= 10.1126/science.236.4808.1519|pmid= 17835731 |bibcode=1987Sci...236.1519K}}</ref> It was not unusual then for marriages to be arranged with girls at a young age. Janaki was from Rajendram, a village close to Marudur ([[Karur district]]) Railway Station. Ramanujan's father did not participate in the marriage ceremony.<ref name="Janaki">{{cite web|title= Ramanujan's wife: Janakiammal (Janaki)|url= https://www.imsc.res.in/~rao/ramanujan/newnow/janaki.pdf|publisher= Institute of Mathematical Sciences|place= Chennai|access-date= 10 November 2012|url-status= dead|archive-url= https://web.archive.org/web/20121224003120/https://www.imsc.res.in/~rao/ramanujan/newnow/janaki.pdf|archive-date= 24 December 2012}}</ref> As was common at that time, Janaki continued to stay at her maternal home for three years after marriage, until she reached puberty. In 1912, she and Ramanujan's mother joined Ramanujan in Madras.<ref name= "express">{{cite news|last1= Janardhanan |first1=Arun|title=A passage to infinity|url= https://indianexpress.com/article/lifestyle/life-style/a-passage-to-infinity-the-untold-story-of-srinivasa-ramanujan/|access-date=7 September 2016|newspaper=Indian Express|date=6 December 2015|url-status=live|archive-url=https://web.archive.org/web/20160905150332/https://indianexpress.com/article/lifestyle/life-style/a-passage-to-infinity-the-untold-story-of-srinivasa-ramanujan/|archive-date=5 September 2016}}</ref>
[[File:Ramanujan seated alone.png|thumb|Ramanujan seated alone]]
After the marriage, Ramanujan developed a [[hydrocele testis]].<ref name="Kanigel-1991-72">>{{harvnb|Kanigel|1991|page=72}}</ref> The condition could be treated with a routine surgical operation that would release the blocked fluid in the scrotal sac, but his family could not afford the operation. In January 1910, a doctor volunteered to do the surgery at no cost.<ref>{{Cite book|last=Ramanujan|first=Srinivasa|editor = P. K. Srinivasan |title= Ramanujan Memorial Number: Letters and Reminiscences |year= 1968|publisher= Muthialpet High School |location=Madras| page=100|volume=1|no-pp=true}}</ref>

After his successful surgery, Ramanujan searched for a job. He stayed at a friend's house while he went from door to door around Madras looking for a clerical position. To make money, he tutored students at Presidency College who were preparing for their Fellow of Arts exam.<ref>{{harvnb|Kanigel|1991|page=73}}</ref>

In late 1910, Ramanujan was sick again. He feared for his health, and told his friend R. Radakrishna Iyer to "hand [his notebooks] over to Professor Singaravelu Mudaliar [the mathematics professor at Pachaiyappa's College] or to the British professor Edward B. Ross, of the [[Madras Christian College]]."<ref>{{harvnb|Kanigel|1991|pages=74–75}}</ref> After Ramanujan recovered and retrieved his notebooks from Iyer, he took a train from Kumbakonam to [[Viluppuram (city)|Villupuram]], a city under French control.<ref>{{Cite book |last=Ranganathan |first=Shiyali Ramamrita |author-link=Shiyali Ramamrita Ranganathan |title=Ramanujan: The Man and the Mathematician |year=1967 |publisher=Asia Publishing House |location=Bombay |url=https://books.google.com/books?id=OuTuAAAAMAAJ |page=23|isbn=9788185273372 }}</ref><ref>Srinivasan (1968), Vol. 1, p. 99.</ref> In 1912, Ramanujan moved with his wife and mother to a house in Saiva Muthaiah Mudali street, [[George Town, Chennai|George Town]], [[Madras]], where they lived for a few months.<ref name="rao">{{cite web|author=Rao, K. Srinivasa|title=Ramanujan's wife Janakiammal (Janaki)|url=https://www.imsc.res.in/~rao/ramanujan/newnow/janaki.pdf|website=IMSC|publisher=Institute of Mathematical Sciences, Chennai|access-date=7 September 2016|url-status=dead|archive-url=https://web.archive.org/web/20170110132908/https://www.imsc.res.in/~rao/ramanujan/newnow/janaki.pdf|archive-date=10 January 2017}}</ref> In May 1913, upon securing a research position at Madras University, Ramanujan moved with his family to [[Triplicane]].<ref>{{cite web|title=About Ramanujan|url=https://ramanujaninstitute.com/about.html|publisher=The Ramanujan Institute|access-date=7 September 2016|url-status=dead|archive-url=https://web.archive.org/web/20161006210739/https://ramanujaninstitute.com/about.html|archive-date=6 October 2016}}</ref>

===Pursuit of career in mathematics===
In 1910, Ramanujan met deputy collector [[V. Ramaswamy Aiyer]], who founded the Indian Mathematical Society.<ref name="Kanigel-1991-77">{{harvnb|Kanigel|1991|page=77}}</ref> Wishing for a job at the revenue department where Aiyer worked, Ramanujan showed him his mathematics notebooks. As Aiyer later recalled:
<blockquote>
I was struck by the extraordinary mathematical results contained in [the notebooks]. I had no mind to smother his genius by an appointment in the lowest rungs of the revenue department.<ref>Srinivasan (1968), Vol. 1, p. 129.</ref>
</blockquote>
Aiyer sent Ramanujan, with letters of introduction, to his mathematician friends in Madras.<ref name="Kanigel-1991-77" /> Some of them looked at his work and gave him letters of introduction to [[R. Ramachandra Rao]], the district collector for [[Nellore]] and the secretary of the Indian Mathematical Society.<ref>Srinivasan (1968), Vol. 1, p. 86.</ref><ref>{{Cite journal|last=Neville |first=Eric Harold |date=January 1921 |title=The Late Srinivasa Ramanujan |journal=[[Nature (journal)|Nature]] |volume=106 |issue=2673 |pages=661–662 |doi=10.1038/106661b0 |bibcode = 1921Natur.106..661N|s2cid=4185656 |url=https://zenodo.org/record/1429646 }}</ref><ref>{{Harvnb|Ranganathan|1967|p=24}}</ref> Rao was impressed by Ramanujan's research but doubted that it was his own work. Ramanujan mentioned a correspondence he had with Professor Saldhana, a notable [[Mumbai|Bombay]] mathematician, in which Saldhana expressed a lack of understanding of his work but concluded that he was not a fraud.<ref name="Kanigel-1991-80">{{harvnb|Kanigel|1991|page=80}}</ref> Ramanujan's friend C. V. Rajagopalachari tried to quell Rao's doubts about Ramanujan's academic integrity. Rao agreed to give him another chance, and listened as Ramanujan discussed [[elliptic integral]]s, [[Gaussian hypergeometric series|hypergeometric series]], and his theory of [[divergent series]], which Rao said ultimately convinced him of Ramanujan's brilliance.<ref name="Kanigel-1991-80" /> When Rao asked him what he wanted, Ramanujan replied that he needed work and financial support. Rao consented and sent him to Madras. He continued his research with Rao's financial aid. With Aiyer's help, Ramanujan had his work published in the ''Journal of the Indian Mathematical Society.<ref>{{harvnb|Kanigel|1991|page=86}}</ref> [[File:K Ananda Rau seated with Ramanujan.jpg|thumb|K Ananda Rau seated with Ramanujan]]
 
One of the first problems he posed in the journal<ref name="Kanigel-1991-unknown-page" /> was to find the value of:

: <math>\sqrt{1+2\sqrt{1+3 \sqrt{1+\cdots}}}</math>

He waited for a solution to be offered in three issues, over six months, but failed to receive any. At the end, Ramanujan supplied an incomplete<ref>{{Cite journal |last=Herschfeld |first=Aaron |date=August 1935 |title=On Infinite Radicals |url=https://www.tandfonline.com/doi/full/10.1080/00029890.1935.11987745 |journal=The American Mathematical Monthly |language=en |volume=42 |issue=7 |pages=419–429 |doi=10.1080/00029890.1935.11987745 |issn=0002-9890}}</ref> solution to the problem himself. On page 105 of his first notebook, he formulated an equation that could be used to solve the infinitely [[nested radical]]s problem.

: <math>x+n+a = \sqrt{ax+(n+a)^2 +x\sqrt{a(x+n)+(n+a)^2+(x+n) \sqrt{\cdots}}}</math>

Using this equation, the answer to the question posed in the ''Journal'' was simply 3, obtained by setting {{math|1=''x'' = 2}}, {{math|1=''n'' = 1}}, and {{math|1=''a'' = 0}}.<ref>{{harvnb|Kanigel|1991|page=87}}</ref> Ramanujan wrote his first formal paper for the ''Journal'' on the properties of [[Bernoulli number]]s. One property he discovered was that the denominators of the fractions of Bernoulli numbers {{OEIS|id=A027642}} are always divisible by six. He also devised a method of calculating {{mvar|B<sub>n</sub>}} based on previous Bernoulli numbers. One of these methods follows:

It will be observed that if ''n'' is even but not equal to zero,
# {{mvar|B<sub>n</sub>}} is a fraction and the numerator of {{math|{{sfrac|''B<sub>n</sub>''|''n''}}}} in its lowest terms is a prime number,
# the denominator of {{mvar|B<sub>n</sub>}} contains each of the factors 2 and 3 once and only once,
# {{math|2<sup>''n''</sup>(2<sup>''n''</sup> − 1){{sfrac|''B<sub>n</sub>''|''n''}}}} is an integer and {{math|2(2<sup>''n''</sup> − 1)''B<sub>n</sub>''}} consequently is an ''odd'' integer.

In his 17-page paper "Some Properties of Bernoulli's Numbers" (1911), Ramanujan gave three proofs, two corollaries and three conjectures.<ref>{{harvnb|Kanigel|1991|page=91}}</ref> His writing initially had many flaws. As ''Journal'' editor M. T. Narayana Iyengar noted:
<blockquote>
Mr. Ramanujan's methods were so terse and novel and his presentation so lacking in clearness and precision, that the ordinary [mathematical reader], unaccustomed to such intellectual gymnastics, could hardly follow him.<ref>{{Cite journal|last=Seshu Iyer |first=P. V. |date=June 1920 |title=The Late Mr. S. Ramanujan, B.A., F.R.S.|journal=Journal of the Indian Mathematical Society |volume=12 |issue=3 |page=83}}</ref></blockquote>
Ramanujan later wrote another paper and also continued to provide problems in the ''Journal''.<ref name="Neville293" /> In early 1912, he got a temporary job in the Madras [[Accountant General]]'s office, with a monthly salary of 20 rupees. He lasted only a few weeks.<ref>Srinivasan (1968), p. 176.</ref> Toward the end of that assignment, he applied for a position under the Chief Accountant of the [[Chennai Port|Madras Port Trust]].

In a letter dated 9 February 1912, Ramanujan wrote:
<blockquote>
Sir,<br />
{{pad|2em}}I understand there is a clerkship vacant in your office, and I beg to apply for the same. I have passed the Matriculation Examination and studied up to the F.A. but was prevented from pursuing my studies further owing to several untoward circumstances. I have, however, been devoting all my time to Mathematics and developing the subject. I can say I am quite confident I can do justice to my work if I am appointed to the post. I therefore beg to request that you will be good enough to confer the appointment on me.<ref>Srinivasan (1968), p. 31.</ref>
</blockquote>

Attached to his application was a recommendation from [[E. W. Middlemast]], a mathematics professor at the [[Presidency College, Chennai|Presidency College]], who wrote that Ramanujan was "a young man of quite exceptional capacity in Mathematics".<ref>Srinivasan (1968), p. 49.</ref> Three weeks after he applied, on 1 March, Ramanujan learned that he had been accepted as a Class III, Grade IV accounting clerk, making 30 rupees per month.<ref>{{harvnb|Kanigel|1991|page=96}}</ref> At his office, Ramanujan easily and quickly completed the work he was given and spent his spare time doing mathematical research. Ramanujan's boss, [[Francis Spring|Sir Francis Spring]], and S. Narayana Iyer, a colleague who was also treasurer of the Indian Mathematical Society, encouraged Ramanujan in his mathematical pursuits.{{sfnp|Berndt|Rankin|2001|p=97}}

===Contacting British mathematicians===
In the spring of 1913, Narayana Iyer, Ramachandra Rao and E. W. Middlemast tried to present Ramanujan's work to British mathematicians. [[Micaiah John Muller Hill|M. J. M. Hill]] of [[University College London]] commented that Ramanujan's papers were riddled with holes.<ref>{{harvnb|Kanigel|1991|page=105}}</ref> He said that although Ramanujan had "a taste for mathematics, and some ability", he lacked the necessary educational background and foundation to be accepted by mathematicians.<ref>Letter from M. J. M. Hill to a C. L. T. Griffith (a former student who sent the request to Hill on Ramanujan's behalf), 28 November 1912.</ref> Although Hill did not offer to take Ramanujan on as a student, he gave thorough and serious professional advice on his work. With the help of friends, Ramanujan drafted letters to leading mathematicians at Cambridge University.<ref>{{harvnb|Kanigel|1991|page=106}}</ref>

The first two professors, [[H. F. Baker]] and [[E. W. Hobson]], returned Ramanujan's papers without comment.<ref>{{harvnb|Kanigel|1991|pages=170–171}}</ref> On 16 January 1913, Ramanujan wrote to [[G. H. Hardy]], whom he knew from studying ''Orders of Infinity'' (1910).<ref>{{Cite book |last=Hardy |first=G. H. (Godfrey Harold) |url=https://www.gutenberg.org/ebooks/38079 |title=Orders of Infinity: The 'Infinitärcalcül' of Paul Du Bois-Reymond |date=1910 |language=English}}</ref><ref>{{Cite journal |last1=Berndt |first1=Bruce C. |last2=Rankin |first2=Robert A. |date=August 2000 |title=The Books Studied by Ramanujan in India |url=https://www.tandfonline.com/doi/full/10.1080/00029890.2000.12005244 |journal=The American Mathematical Monthly |language=en |volume=107 |issue=7 |pages=595–601 |doi=10.1080/00029890.2000.12005244 |issn=0002-9890}}</ref> Coming from an unknown mathematician, the nine pages of mathematics made Hardy initially view Ramanujan's manuscripts as a possible fraud.<ref>{{Cite book|last=Snow|first=C. P. |title= Variety of Men |year= 1966|publisher= [[Charles Scribner's Sons]] |location=New York| pages=30–31}}</ref> Hardy recognised some of Ramanujan's formulae but others "seemed scarcely possible to believe".<ref name="Hardy1920">{{Cite journal|last=Hardy |first=G. H.|author-link=G. H. Hardy |year=1920 |title=Obituary, S. Ramanujan |journal=Nature|volume=105 |issue= 7|pages=494–495 |doi=10.1038/105494a0 |bibcode = 1920Natur.105..494H|s2cid=4174904|url=https://zenodo.org/record/1429640 |doi-access=free}}</ref>{{rp|494}} One of the theorems Hardy found amazing was on the bottom of page three (valid for {{math|0 < ''a'' < ''b'' + {{sfrac|1|2}}}}):

: <math>
\int\limits_0^\infty \frac{1+\dfrac{x^2}{(b+1)^2}}{1+\dfrac{x^2}{a^2}} \times\frac{1+\dfrac{x^2}{(b+2)^2}}{1+\dfrac{x^2}{(a+1)^2}}\times\cdots\,dx
= \frac{\sqrt \pi}{2} \times\frac{\Gamma\left(a+\frac12\right) \Gamma(b+1) \Gamma(b-a+1)}{\Gamma(a)\Gamma\left(b+\frac12\right)\Gamma\left(b-a + \frac12 \right)}.
</math>

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

: <math>1 - 5\left(\frac12\right)^3 + 9\left(\frac{1\times3}{2\times4}\right)^3 - 13\left(\frac{1\times3\times5}{2\times4\times6}\right)^3 + \cdots = \frac{2}{\pi}</math>

: <math>1 + 9\left(\frac14\right)^4 + 17\left(\frac{1\times5}{4\times8}\right)^4 + 25\left(\frac{1\times5\times9}{4\times8\times12}\right)^4 + \cdots = \frac{2\sqrt 2}{\sqrt\pi\,\Gamma^2\left(\frac34\right)}.</math>

The first result had already been determined by [[Gustav Conrad Bauer|G. Bauer]] in 1859. The second was new to Hardy, and was derived from a class of functions called [[hypergeometric series]], which had first been researched by Euler and Gauss. Hardy found these results "much more intriguing" than Gauss's work on integrals.<ref name="Kanigel-1991-167">{{harvnb|Kanigel|1991|page=167}}</ref> After seeing [[Rogers–Ramanujan continued fraction|Ramanujan's theorems on continued fractions]] on the last page of the manuscripts, Hardy said the theorems "defeated me completely; I had never seen anything in the least like them before",<ref name="Kanigel-1991-168">{{harvnb|Kanigel|1991|page=168}}</ref> and that they "must be true, because, if they were not true, no one would have the imagination to invent them".<ref name="Kanigel-1991-168" /> Hardy asked a colleague, [[John Edensor Littlewood|J. E. Littlewood]], to take a look at the papers. Littlewood was amazed by Ramanujan's genius. After discussing the papers with Littlewood, Hardy concluded that the letters were "certainly the most remarkable I have received" and that Ramanujan was "a mathematician of the highest quality, a man of altogether exceptional originality and power".<ref name="Hardy1920" />{{rp|494–495}} One colleague, [[Eric Harold Neville|E. H. Neville]], later remarked that "No one who was in the mathematical circles in Cambridge at that time can forget the sensation caused by this letter... not one [theorem] could have been set in the most advanced mathematical examination in the world".<ref name="Neville293">{{Cite journal|last=Neville |first=Eric Harold |date=1942 |title=Srinivasa Ramanujan |journal=Nature |volume=149 |issue=3776 |pages=292–293 |doi=10.1038/149292a0 |bibcode = 1942Natur.149..292N |doi-access=free }}</ref>

On 8 February 1913, Hardy wrote Ramanujan a letter expressing interest in his work, adding that it was "essential that I should see proofs of some of your assertions".<ref>Letter, Hardy to Ramanujan, 8 February 1913.</ref> Before his letter arrived in Madras during the third week of February, Hardy contacted the Indian 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 overseas trip.<ref>Letter, Ramanujan to Hardy, 22 January 1914.</ref> In accordance with his Brahmin upbringing, Ramanujan refused to leave his country to "[[Kala pani (taboo)|go to a foreign land]]", and his parents were also opposed for the same reason.<ref>{{harvnb|Kanigel|1991|page=185}}</ref> Meanwhile, he sent Hardy a letter packed with theorems, writing, "I have found a friend in you who views my labour sympathetically."<ref>Letter, Ramanujan to Hardy, 27 February 1913, [[Cambridge University Library]].</ref>

To supplement Hardy's endorsement, [[Gilbert Walker (physicist)|Gilbert Walker]], a former mathematical lecturer at [[Trinity College, Cambridge]], looked at Ramanujan's work and expressed amazement, urging the young man to spend time at Cambridge.<ref>{{harvnb|Kanigel|1991|page=175}}</ref> As a result of Walker's endorsement, B. Hanumantha Rao, a mathematics professor at an engineering college, invited Ramanujan's colleague Narayana Iyer to a meeting of the Board of Studies in Mathematics to discuss "what we can do for S. Ramanujan".<ref>{{Cite book|last=Ram|first=Suresh |title= Srinivasa Ramanujan |year= 1972|publisher= National Book Trust |location=New Delhi| page=29}}</ref> The board agreed to grant Ramanujan a monthly research scholarship of 75 rupees for the next two years at the [[University of Madras]].<ref>{{Harvnb|Ranganathan|1967|pp=30–31}}</ref>

While he was engaged as a research student, Ramanujan continued to submit papers to the ''Journal of the Indian Mathematical Society.'' In one instance, Iyer submitted some of Ramanujan's theorems on summation of series to the journal, adding, "The following theorem is due to S. Ramanujan, the mathematics student of Madras University." Later in November, British Professor [[Edward Burns Ross|Edward B. Ross]] of [[Madras Christian College]], whom Ramanujan had met a few years before, stormed into his class one day with his eyes glowing, asking his students, "Does Ramanujan know Polish?" The reason was that in one paper, Ramanujan had anticipated the work of a Polish mathematician whose paper had just arrived in the day's mail.<ref>{{Harvnb|Ranganathan|1967|p=12}}</ref> In his quarterly papers, Ramanujan drew up theorems to make definite integrals more easily solvable. Working off Giuliano Frullani's 1821 integral theorem, Ramanujan formulated generalisations that could be made to evaluate formerly unyielding integrals.<ref>{{harvnb|Kanigel|1991|page=183}}</ref>

Hardy's correspondence with Ramanujan soured after Ramanujan refused to come to England. Hardy enlisted a colleague lecturing in Madras, E. H. Neville, to mentor and bring Ramanujan to England.<ref>{{harvnb|Kanigel|1991|page=184}}</ref> Neville asked Ramanujan why he would not go to Cambridge. Ramanujan apparently had now accepted the proposal; Neville said, "Ramanujan needed no converting" and "his parents' opposition had been withdrawn".<ref name="Neville293"/> Apparently, Ramanujan's mother had a vivid dream in which Ramanujan was surrounded by Europeans, and the family goddess, [[Namagiri Thayar|the deity of Namagiri]], commanded her "to stand no longer between her son and the fulfilment of his life's purpose".<ref name="Neville293"/> On 17 March 1914, Ramanujan travelled to England by ship,<ref>{{cite web |title=A (very) Brief History of Srinivasa Ramanujan |via = [[YouTube]]| date=31 January 2021 |url=https://www.youtube.com/watch?v=GyFYF27xFMU| archive-url=https://ghostarchive.org/varchive/youtube/20211211/GyFYF27xFMU| archive-date=2021-12-11 | url-status=live}}{{cbignore}}</ref> leaving his wife to stay with his parents in India.{{sfnp|Berndt|Rankin|2001|pp=83–84}}

==Life in England==
[[File:RamanujanCambridge.jpg|thumb|upright=1.3|Ramanujan (centre) and his colleague [[G. H. Hardy]] (rightmost), with other scientists, outside the [[Senate House, Cambridge]], c.1914–19]]
[[File:Whewell's Court, Trinity College, Cambridge.jpg|thumb|upright|Whewell's Court, [[Trinity College, Cambridge]]]]

Ramanujan departed from Madras aboard the S.S. ''Nevasa'' on 17 March 1914.<ref>{{harvnb|Kanigel|1991|page=196}}</ref><ref>{{cite book |title=Lloyd's Register of Shipping 1930 |date=1930 |url=https://plimsoll.southampton.gov.uk/shipdata/pdfs/30/30b0845.pdf |accessdate=1 September 2020 |quote=Nevasa 1913}}</ref>  When he disembarked in London on 14 April, Neville was waiting for him with a car. Four days later, Neville took him to his house on Chesterton Road in Cambridge. Ramanujan immediately began his work with Littlewood and Hardy. After six weeks, Ramanujan moved out of Neville's house and took up residence on Whewell's Court, a five-minute walk from Hardy's room.<ref>{{harvnb|Kanigel|1991|page=202}}</ref> [[File:Ramanujan's "Master Theorem" page.jpg|thumb|Ramanujan's "Master Theorem" page]]

[[G. H. Hardy|Hardy]] and [[J. E. Littlewood|Littlewood]] began to look at Ramanujan's notebooks. Hardy had already received 120 theorems from Ramanujan in the first two letters, but there were many more results and theorems in the notebooks. Hardy saw that some were wrong, others had already been discovered, and the rest were new breakthroughs.<ref>{{Cite book|last=Hardy |first=G. H. |title= Ramanujan|year= 1940 |publisher= [[Cambridge University Press]] |location=Cambridge| page=10}}</ref> Ramanujan left a deep impression on Hardy and Littlewood. Littlewood commented, "I can believe that he's at least a [[Carl Gustav Jacob Jacobi|Jacobi]]",<ref>Letter, Littlewood to Hardy, early March 1913.</ref> while Hardy said he "can compare him only with [[Leonhard Euler|Euler]] or Jacobi."<ref>{{Cite book|last=Hardy |first=G. H. |title= Collected Papers of G. H. Hardy|year= 1979 |publisher= [[Oxford University Press|Clarendon Press]] |location=Oxford, England| page=720|volume=7|no-pp=true}}</ref>

Ramanujan spent nearly five years in [[Cambridge]] collaborating with Hardy and Littlewood, and published part of his findings there. Hardy and Ramanujan had highly contrasting personalities. Their collaboration was a clash of different cultures, beliefs, and working styles. In the previous few decades, the [[foundations of mathematics]] had come into question and the need for [[mathematical rigor|mathematically rigorous]] proofs was recognised. Hardy was an atheist and an apostle of proof and mathematical rigour, whereas Ramanujan was a deeply religious man who relied very strongly on his intuition and insights. Hardy tried his best to fill the gaps in Ramanujan's education and to mentor him in the need for formal proofs to support his results, without hindering his inspiration—a conflict that neither found easy.

Ramanujan was awarded a ''Bachelor of Arts by Research'' degree<ref>The Cambridge University Reporter, of 18 March 1916, reports: ''Bachelors designate in Arts,'' Srinivasa Ramanujan ''(Research Student),'' Trin''.''

A clear photographic image of said document can be viewed on the following YouTube video at the specified timestamp:

https://www.youtube.com/watch?v=uhNGCn_3hmc&t=1636</ref><ref>{{Cite web|title=The Maths PhD in the UK: Notes on its History|url=http://www.economics.soton.ac.uk/staff/aldrich/Doc1.htm|access-date=2020-08-09|website=www.economics.soton.ac.uk}}</ref> (the predecessor of the PhD degree) in March 1916 for his work on [[highly composite number]]s, sections of the first part of which had been published the preceding year in the ''[[Proceedings of the London Mathematical Society]].'' The paper was more than 50 pages long and proved various properties of such numbers. Hardy disliked this topic area but remarked that though it engaged with what he called the 'backwater of mathematics', in it Ramanujan displayed 'extraordinary mastery over the algebra of inequalities'.<ref>Jean-Louis Nicolas, Guy Robin (eds.), [http://math.univ-lyon1.fr/~nicolas/ramanujanNR.pdf ''Highly Composite Numbers by Srinivasa Ramanujan,''] The Ramanujan Journal 1997 1, 119–153, p.121</ref>

On 6 December 1917, Ramanujan was elected to the London Mathematical Society. On 2 May 1918, he was elected a [[Fellow of the Royal Society]],<ref name="FRS-Repository">{{cite web |last1=Embleton |first1=Ellen |title=Revisiting Ramanujan |url=https://blogs.royalsociety.org/history-of-science/2018/10/02/revisiting-ramanujan/ |website=The Royal Society |access-date=16 February 2020 |date=2 October 2018 |archive-date=16 February 2020 |archive-url=https://web.archive.org/web/20200216230325/https://blogs.royalsociety.org/history-of-science/2018/10/02/revisiting-ramanujan/ |url-status=dead }}</ref> the second Indian admitted, after [[Ardaseer Cursetjee]] in 1841. At age 31, Ramanujan was one of the youngest Fellows in the Royal Society's history. He was elected "for his investigation in [[elliptic function]]s and the Theory of Numbers." On 13 October 1918, he was the first Indian to be elected a [[Trinity College, Cambridge#Notable fellows and alumni|Fellow of Trinity College, Cambridge]].<ref>{{harvnb|Kanigel|1991|pages=299–300}}</ref>

==Illness and death==
Ramanujan had numerous health problems throughout his life. His health worsened in England; possibly he was also less resilient due to the difficulty of keeping to the strict dietary requirements of his religion there and because of [[Rationing in the United Kingdom#First World War 1914–1918|wartime rationing in 1914–18]]. He was diagnosed with [[tuberculosis]] and a severe [[vitamin]] deficiency, and confined to a [[sanatorium]]. He attempted suicide in late 1917 or early 1918 by jumping on the tracks of a London underground station. [[Scotland Yard]] arrested him for attempting suicide (which was a crime), but released him after Hardy intervened.<ref>S. Chandrasekhar, ''An Incident in the Life of S. Ramanujan F.R.S. Conversations with G.H. Hardy FRS & J.E. Littlewood FRS and their Sequel''. Papers deposited with the Royal Society (18/8/1977). Reprinted in Berndt, Bruce C., and Robert Alexander Rankin. ''Ramanujan: essays and surveys''. Providence: American Mathematical Society, 2001.</ref><ref name=":1" /> In 1919, Ramanujan returned to [[Kumbakonam]], [[Madras Presidency]], where he died in 1920 aged 32. After his death, his brother Tirunarayanan compiled Ramanujan's remaining handwritten notes, consisting of formulae on singular moduli, hypergeometric series and continued fractions.<ref name="express"/> In his last days, though in severe pain, "he continued doing his mathematics filling sheet after sheet with numbers", Janaki Ammal recounts.<ref>Ramaseshan, S. "Srinivasa Ramanujan." (1990). CURRENT SCIENCE, VOL. 59, NO. 24, 25 DECEMBER 1990

Lecture delivered at the Ramanujan Centennial International Conference (15-18 December 1987) at Kumbakonam.</ref>

Ramanujan's widow, [[Smt.]] Janaki Ammal, moved to [[Bombay]]. In 1931, she returned to Madras and settled in [[Triplicane]], where she supported herself on a pension from Madras University and income from tailoring. In 1950, she adopted a son, W. Narayanan, who eventually became an officer of the [[State Bank of India]] and raised a family. In her later years, she was granted a lifetime pension from Ramanujan's former employer, the Madras Port Trust, and pensions from, among others, the [[Indian National Science Academy]] and the state governments of [[Tamil Nadu]], [[Andhra Pradesh]] and [[West Bengal]]. She continued to cherish Ramanujan's memory, and was active in efforts to increase his public recognition; prominent mathematicians, including George Andrews, [[Bruce C. Berndt]] and [[Béla Bollobás]] made it a point to visit her while in India. She died at her Triplicane residence in 1994.<ref name="Janaki"/><ref name="express"/>

A 1994 analysis of Ramanujan's medical records and symptoms by D. A. B. Young<ref name=":1">{{cite journal|doi=10.1098/rsnr.1994.0009 |author=Young, D. A. B. |year=1994|title=Ramanujan's illness|journal= Notes and Records of the Royal Society of London|volume= 48|issue=1|pages= 107–119|pmid=11615274 |s2cid=33416179 }}</ref> concluded that his medical [[symptom]]s—including his past relapses, fevers, and hepatic conditions—were much closer to those of hepatic [[amoebiasis]], an illness then widespread in Madras, than of tuberculosis. He had two episodes of [[dysentery]] before he left India. When not properly treated, amoebic dysentery can lie dormant for years and lead to hepatic amoebiasis, whose diagnosis was not then well established.<ref name="lostnotebook">{{cite web|url=https://www.las.illinois.edu/alumni/magazine/articles/2006/lostnotebook/|title=Raiders of the Lost Notebook|access-date=11 January 2014|last=Peterson|first=Doug|publisher=[[UIUC College of Liberal Arts and Sciences]]|url-status=dead|archive-url=https://web.archive.org/web/20140112022743/https://www.las.illinois.edu/alumni/magazine/articles/2006/lostnotebook/|archive-date=12 January 2014}}</ref> At the time, if properly diagnosed, amoebiasis was a treatable and often curable disease;<ref name="lostnotebook"/><ref>{{cite journal |author=Gunn, J. W. C. |author2=Savage, B. |year=1919|title=Report on the treatment of ''Entamoeba histolytica'' infections|journal= Journal of the Royal Army Medical Corps|volume=33|issue=5|pages= 418–426}}</ref> British soldiers who contracted it during the First World War were being successfully cured of amoebiasis around the time Ramanujan left England.<ref>{{Cite journal | last1 =Langley | first1 =George J.  |jstor =20429465 | title =The Difficulties in Diagnosis And Treatment of Hepatic Abscess | journal =[[British Medical Journal]]| volume =2  | issue =3182  | pages =1073–1074 | date =24 December 1921 | pmid =  20770524| pmc = 2339657| doi =10.1136/bmj.2.3182.1073 }}</ref>

==Personality and spiritual life==
{{Quote box
 | quote  = While asleep, I had an unusual experience. There was a red screen formed by flowing blood, as it were. I was observing it. Suddenly a hand began to write on the screen. I became all attention. That hand wrote a number of elliptic integrals. They stuck to my mind. As soon as I woke up, I committed them to writing.
 | source = —Srinivasa Ramanujan<ref>{{cite book |last1=Baaquie |first1=Belal E. |last2=Willeboordse |first2=Frederick H. |title=Exploring Integrated Science |date=1 December 2009 |publisher=CRC Press |isbn=978-1-4200-8794-9 |page=38 |url=https://books.google.com/books?id=WSf1FFOTIYEC&pg=PA38 |language=en}}</ref>
 | width  = 30%
 | align  = right
}}

Ramanujan has been described as a person of a somewhat shy and quiet disposition, a dignified man with pleasant manners.<ref>{{cite web |url = https://www.imsc.res.in/~rao/ramanujan/newnow/pcm5.htm |title = Ramanujan's Personality |url-status = dead |archive-url = https://web.archive.org/web/20070927045011/https://www.imsc.res.in/~rao/ramanujan/newnow/pcm5.htm |archive-date = 27 September 2007 |df = dmy-all |access-date = 23 June 2018 }}</ref> He lived a simple life at Cambridge.<ref>{{harvnb|Kanigel|1991|pages=234, 241}}</ref> Ramanujan's first Indian biographers describe him as a rigorously [[Orthodox Hinduism|orthodox Hindu]]. He credited his acumen to his [[Kuladevata|family goddess]], [[Namagiri Thayar]] (Goddess [[Lakshmi|Mahalakshmi]]) of [[Namakkal]].  He looked to her for inspiration in his work<ref>{{harvnb|Kanigel|1991|page=36}}</ref> and said he dreamed of blood drops that symbolised her consort, [[Narasimha]]. Later he had visions of scrolls of complex mathematical content unfolding before his eyes.<ref name="Kanigel-1991-281">{{harvnb|Kanigel|1991|page=281}}</ref> He often said, "An equation for me has no meaning unless it expresses a thought of God."<ref>{{Cite journal|title= Less Proof, More Truth |journal= New Scientist |issue=2614 |date= 28 July 2007 |page=49 | last =Chaitin | first = Gregory |doi= 10.1016/S0262-4079(07)61908-3 |url= https://philpapers.org/rec/CHALPM}}</ref>

Hardy cites Ramanujan as remarking that all religions seemed equally true to him.<ref name="Kanigel-1991-283">{{harvnb|Kanigel|1991|page=283}}</ref> Hardy further argued that Ramanujan's religious belief had been romanticised by Westerners and overstated—in reference to his belief, not practice—by Indian biographers.  At the same time, he remarked on Ramanujan's strict [[vegetarian]]ism.<ref name= "berndt">{{Harvnb|Berndt|Rankin|2001|p=47}}</ref>

Similarly, in an interview with Frontline, Berndt said, "Many people falsely promulgate mystical powers to Ramanujan's mathematical thinking. It is not true. He has meticulously recorded every result in his three notebooks," further speculating that Ramanujan worked out intermediate results on slate that he could not afford the paper to record more permanently.<ref name= "frontline" />

Berndt reported that Janaki said in 1984 that Ramanujan spent so much of his time on mathematics that he did not go to the temple, that she and her mother often fed him because he had no time to eat, and that most of the religious stories attributed to him originated with others. However, his [[orthopraxy]] was not in doubt.<ref>{{Cite journal |last=Berndt |first=Bruce C. |date=1989 |title=Srinivasa Ramanujan |url=https://www.jstor.org/stable/41211668 |journal=The American Scholar |volume=58 |issue=2 |pages=234–244 |jstor=41211668 |issn=0003-0937}}</ref>

==Mathematical achievements==
In mathematics, there is a distinction between insight and formulating or working through a proof. Ramanujan proposed an abundance of formulae that could be investigated later in depth. [[G.&nbsp;H. Hardy]] said that Ramanujan's discoveries are unusually rich and that there is often more to them than initially meets the eye. As a byproduct of his work, new directions of research were opened up. Examples of the most intriguing of these formulae include infinite [[Series (mathematics)|series]] for [[pi|{{pi}}]], one of which is given below:

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

This result is based on the negative [[fundamental discriminant]] {{nowrap|1={{mvar|d}} = −4 × 58 = −232}} with class number {{nowrap|1={{math|''h''(''d'')}} = 2}}. Further, {{nowrap|1=26390 = 5 × 7 × 13 × 58}} and {{nowrap|1=16 × 9801 = 396<sup>2</sup>}}, which is related to the fact that

:<math display="inline"> e^{\pi\sqrt{58}} = 396^4 - 104.000000177\dots. </math>

This might be compared to [[Heegner number]]s, which have [[Ideal class group|class number]] 1 and yield similar formulae.

Ramanujan's series for {{pi}} converges extraordinarily rapidly and forms the basis of some of the fastest algorithms used to calculate {{pi}}. Truncating the sum to the first term also gives the approximation {{sfrac|9801{{sqrt|2}}|4412}} for {{pi}}, which is correct to six decimal places; truncating it to the first two terms gives a value correct to 14 decimal places {{crossreference|(see also the more general [[Ramanujan–Sato series]])}}.

One of Ramanujan's remarkable capabilities was the rapid solution of problems, illustrated by the following anecdote about an incident in which [[Mahalanobis|P. C. Mahalanobis]] posed a problem:

{{blockquote|Imagine that you are on a street with houses marked 1 through {{mvar|n}}. There is a house in between ({{mvar|x}}) such that the sum of the house numbers to the left of it equals the sum of the house numbers to its right. If {{mvar|n}} is between 50 and 500, what are {{mvar|n}} and {{mvar|x}}?' This is a bivariate problem with multiple solutions. Ramanujan thought about it and gave the answer with a twist: He gave a [[continued fraction]]. The unusual part was that it was the solution to the whole class of problems.  Mahalanobis was astounded and asked how he did it. 'It is simple. The minute I heard the problem, I knew that the answer was a continued fraction. Which continued fraction, I asked myself. Then the answer came to my mind', Ramanujan replied."<ref>{{Harvnb|Ranganathan|1967|p=82}}</ref><ref>{{Cite book|title=Statistics and truth: putting chance to work|year=1997|publisher=World Scientific|url=https://books.google.com/books?id=jqWd4oe3iwIC&q=%22Which+continued+fraction%22&pg=PA185|author=Calyampudi Radhakrishna Rao|access-date=7 June 2010|page=185|isbn=978-981-02-3111-8}}</ref>}}

His intuition also led him to derive some previously unknown [[identity (mathematics)|identities]], such as

: <math>
\begin{align}
& \left ( 1+2\sum_{n=1}^\infty \frac{\cos(n\theta)}{\cosh(n\pi)} \right )^{-2} + \left (1+2\sum_{n=1}^\infty \frac{\cosh(n\theta)}{\cosh(n\pi)} \right )^{-2} \\[6pt]
= {} & \frac {2 \Gamma^4 \bigl( \frac34 \bigr)} \pi = \frac{8\pi^3}{\Gamma^4 \bigl( \frac14 \bigr)}
\end{align}
</math>

for all {{math|''θ''}} such that <math>|\Re (\theta)|<\pi</math> and <math>|\Im (\theta)|<\pi</math>, where {{math|Γ(''z'')}} is the [[gamma function]], and related to a special value of the [[Dedekind eta function#Special values|Dedekind eta function]]. Expanding into series of powers and equating coefficients of {{math|''θ''<sup>0</sup>}}, {{math|''θ''<sup>4</sup>}}, and {{math|''θ''<sup>8</sup>}} gives some deep identities for the [[hyperbolic secant]].

In 1918, Hardy and Ramanujan studied the [[Partition function (number theory)|partition function]] {{math|''P''(''n'')}} extensively. They gave a non-convergent asymptotic series that permits exact computation of the number of partitions of an integer. In 1937, [[Hans Rademacher]] refined their formula to find an exact convergent series solution to this problem. Ramanujan and Hardy's work in this area gave rise to a powerful new method for finding asymptotic formulae called the [[Hardy–Littlewood circle method|circle method]].<ref>{{cite conference
 | last = Iwaniec | first = Henryk | author-link = Henryk Iwaniec
 | contribution = The circle method and the Fourier coefficients of modular forms
 | mr = 1441326
 | pages = 47–55
 | publisher = Tata Inst. Fund. Res., Bombay
 | series = Tata Inst. Fund. Res. Stud. Math.
 | title = Number theory and related topics (Bombay, 1988)
 | volume = 12
 | year = 1989}}</ref>

In the last year of his life, Ramanujan discovered [[mock theta function]]s.<ref name="Fox">{{cite news | url=https://www.foxnews.com/science/100-year-old-deathbed-dreams-of-mathematician-proved-true/ | work=Fox News | title=100-Year-Old Deathbed Dreams of Mathematician Proved True | date=28 December 2012 | url-status=live | archive-url=https://web.archive.org/web/20130107003536/https://www.foxnews.com/science/2012/12/28/mathematician-century-old-secrets-unlocked/ | archive-date=7 January 2013 | df=dmy-all }}</ref> For many years, these functions were a mystery, but they are now known to be the holomorphic parts of harmonic weak [[Maass forms]].

===The Ramanujan conjecture===
{{Main|Ramanujan–Petersson conjecture}}

Although there are numerous statements that could have borne the name ''Ramanujan conjecture,'' one was highly influential in later work. In particular, the connection of this conjecture with conjectures of [[André Weil]] in algebraic geometry opened up new areas of research. That [[Ramanujan conjecture]] is an assertion on the size of the [[Ramanujan tau function|tau-function]], which has a generating function as the discriminant modular form Δ(''q''), a typical [[cusp form]] in the theory of [[modular forms]]. It was finally proven in 1973, as a consequence of [[Pierre Deligne]]'s proof of the [[Weil conjectures]]. The reduction step involved is complicated. Deligne won a [[Fields Medal]] in 1978 for that work.<ref name=ono/><ref>{{Cite journal|date=December 9, 2019|last=Winnie|first=Wen-Ching|title=The Ramanujan conjecture and its applications|journal=Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences|volume=378 |issue=2163 |doi=10.1098/rsta.2018.0441 |pmid=31813366 |pmc=6939229 }}</ref>

In his paper "On certain arithmetical functions", Ramanujan defined the so-called delta-function, whose coefficients are called {{math|''τ''(''n'')}} (the [[Ramanujan tau function]]).<ref>{{cite journal |last= Ramanujan |first= Srinivasa |title= On certain arithmetical functions |journal= Transactions of the Cambridge Philosophical Society |volume=XXII |number= 9 |year= 1916 |url= https://ramanujan.sirinudi.org/Volumes/published/ram18.pdf |access-date= 15 May 2016 |url-status=dead |archive-url= https://web.archive.org/web/20160611175341/https://ramanujan.sirinudi.org/Volumes/published/ram18.pdf |archive-date= 11 June 2016 |df= dmy-all }} The tau function is discussed in pages 194–197.</ref> He proved many congruences for these numbers, such as {{math|''τ''(''p'') ≡ 1 + ''p''<sup>11</sup> mod 691}} for primes {{mvar|p}}. This congruence (and others like it that Ramanujan proved) inspired [[Jean-Pierre Serre]] (1954 Fields Medalist) to conjecture that there is a theory of [[Galois module|Galois representations]] that "explains" these congruences and more generally all modular forms. {{math|Δ(''z'')}} is the first example of a modular form to be studied in this way. Deligne (in his Fields Medal-winning work) proved Serre's conjecture. The proof of [[Fermat's Last Theorem]] proceeds by first reinterpreting [[elliptic curve]]s and modular forms in terms of these Galois representations. Without this theory, there would be no proof of Fermat's Last Theorem.<ref>{{cite book|last1= Ono |first1= Ken |author-link1= Ken Ono |last2= Aczel|first2= Amir D. |author-link2= Amir Aczel |title= My Search for Ramanujan: How I Learned to Count |date= 13 April 2016 |publisher= [[Springer Science+Business Media|Springer]] |pages= 236–237 |isbn= 978-3319255668 |url= https://books.google.com/books?id=JZkFDAAAQBAJ&q=%22ideas+that+were+critical+to+the+proof+of+Fermat%27s+last+theorem%22&pg=PA236 |quote= ideas that were critical to the proof of Fermat's last theorem}}</ref>

===Ramanujan's notebooks===
{{Further|Ramanujan's lost notebook}}

While still in Madras, Ramanujan recorded the bulk of his results in four notebooks of [[loose leaf|looseleaf]] paper. They were mostly written up without any derivations. This is probably the origin of the misapprehension that Ramanujan was unable to prove his results and simply thought up the final result directly. Mathematician [[Bruce C. Berndt]], in his review of these notebooks and Ramanujan's work, says that Ramanujan most certainly was able to prove most of his results, but chose not to record the proofs in his notes.

This may have been for any number of reasons. Since paper was very expensive, Ramanujan did most of his work and perhaps his proofs on [[slate (writing)|slate]], after which he transferred the final results to paper. At the time, slates were commonly used by mathematics students in the [[Madras Presidency]]. He was also quite likely to have been influenced by the style of [[G. S. Carr]]'s book, which stated results without proofs. It is also possible that Ramanujan considered his work to be for his personal interest alone and therefore recorded only the results.<ref name="Bruce Berndt on Ramanujan">{{Cite book|isbn = 978-0387949413 |title = Ramanujans Notebooks|last1 = Berndt|first1 = Bruce C.|date = 12 December 1997| publisher=Springer }}</ref>

The first notebook has 351 pages with 16 somewhat organised chapters and some unorganised material. The second has 256 pages in 21 chapters and 100 unorganised pages, and the third 33 unorganised pages. The results in his notebooks inspired numerous papers by later mathematicians trying to prove what he had found. Hardy himself wrote papers exploring material from Ramanujan's work, as did [[G. N. Watson]], [[Bertram Martin Wilson|B. M. Wilson]], and Bruce Berndt.<ref name="Bruce Berndt on Ramanujan"/>

In 1976, [[George Andrews (mathematician)|George Andrews]] rediscovered a fourth notebook with 87 unorganised pages, the so-called [[Ramanujan's lost notebook|"lost notebook"]].<ref name="lostnotebook"/>

==Hardy–Ramanujan number 1729==
{{Main|1729 (number)}}
The number 1729 is known as the Hardy–Ramanujan number after a famous visit by Hardy to see Ramanujan at a hospital. In Hardy's words:<ref>{{cite web |url=https://www-gap.dcs.st-and.ac.uk/~history/Quotations/Hardy.html |title=Quotations by Hardy |publisher=Gap.dcs.st-and.ac.uk |access-date=20 November 2012 |url-status=dead |archive-url=https://web.archive.org/web/20120716185939/https://www-gap.dcs.st-and.ac.uk/~history/Quotations/Hardy.html |archive-date=16 July 2012  }}</ref>

<blockquote>I remember once going to see him when he was ill at [[Putney]]. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a [[Interesting number paradox|dull one]], and that I hoped it was not an unfavorable omen. "No", he replied, "it is a very interesting number; it is the smallest number expressible as the [[sum of two cubes]] in two different ways."</blockquote>

Immediately before this anecdote, Hardy quoted Littlewood as saying, "Every positive integer was one of [Ramanujan's] personal friends."<ref>{{cite journal|url=https://www.numbertheory.org/obituaries/LMS/ramanujan/page18.html |title=Obituary Notices: Srinivasa Ramanujan |last=Hardy |first=G. H. |journal=Proceedings of the London Mathematical Society |volume=19 |page=lvii |url-status=live |archive-url=https://web.archive.org/web/20160305005434/https://www.numbertheory.org/obituaries/LMS/ramanujan/page18.html |archive-date=5 March 2016  }}</ref>

The two different ways are:

:<math>1729=1^3+12^3=9^3+10^3.</math>

Generalisations of this idea have created the notion of "[[taxicab number]]s".

==Mathematicians' views of Ramanujan==
{{Quote box
 | quote  = "Of course, we're always hoping. That's one reason I always read letters that come in from obscure places and are written in an illegible scrawl. I always hope it might be from another Ramanujan."

 | source = —Freeman Dyson on how another such genius might appear anywhere<ref name="Ross 2017 u486">{{cite web | last=Ross | first=Greg | title=Podcast Episode 140: Ramanujan | website=Futility Closet | date=2017-02-06 | url=https://www.futilitycloset.com/2017/02/06/podcast-episode-140-ramanujan/ | access-date=2023-12-24}}</ref>
 | width  = 30%
 | align  = right
}}

In his obituary of Ramanujan, written for ''Nature'' in 1920, Hardy observed that Ramanujan's work primarily involved fields less known even among other pure mathematicians, concluding:
{{blockquote|His insight into formulae was quite amazing, and altogether beyond anything I have met with in any European mathematician. It is perhaps useless to speculate as to his history had he been introduced to modern ideas and methods at sixteen instead of at twenty-six. It is not extravagant to suppose that he might have become the greatest mathematician of his time. What he actually did is wonderful enough… when the researches which his work has suggested have been completed, it will probably seem a good deal more wonderful than it does to-day.<ref name="Hardy1920"/>}}

Hardy further said:<ref>{{Cite book |title=The world of mathematics|date=2000|publisher=Dover Publications|editor=James R. Newman|isbn=978-0-486-41153-8|location=Mineola, NY |pages=373–4|oclc=43555029}}</ref>

{{blockquote|He combined a power of generalisation, a feeling for form, and a capacity for rapid modification of his hypotheses, that were often really startling, and made him, in his own peculiar field, without a rival in his day. The limitations of his knowledge were as startling as its profundity. Here was a man who could work out [[modular equation]]s and theorems... to orders unheard of, whose mastery of continued fractions was... beyond that of any mathematician in the world, who had found for himself the functional equation of the [[Riemann zeta function|zeta function]] and the dominant terms of many of the most famous problems in the analytic theory of numbers; and yet he had never heard of a [[doubly periodic function]] or of [[Cauchy's integral theorem|Cauchy's theorem]], and had indeed but the vaguest idea of what a function of a [[complex variable]] was..."}}

As an example, Hardy commented on 15 theorems in the first letter. Of those, the first 13 are correct and insightful, the 14th is incorrect but insightful, and the 15th is correct but misleading.

(14): The coefficient of <math>x^n</math> in <math>\left(1-2 x+2 x^4-2 x^9+\cdots\right)^{-1}</math> is the integer nearest to<math display="block">
\frac{1}{4 n}\left(\cosh (\pi \sqrt{n})-\frac{\sinh (\pi \sqrt{ n})}{\pi \sqrt{n}}\right) .
</math>This "was one of the most fruitful he ever made, since it ended by leading us to all our joint work on partitions".<ref name=":0" />

When asked about the methods Ramanujan used to arrive at his solutions, Hardy said they were "arrived at by a process of mingled argument, intuition, and induction, of which he was entirely unable to give any coherent account."<ref name="cola1">[https://www.usna.edu/Users/math/meh/ramanujan.html Srinivasa Ramanujan] {{webarchive|url=https://web.archive.org/web/20050325041218/https://www.usna.edu/Users/math/meh/ramanujan.html |date=25 March 2005 }}. Retrieved 2 December 2010.</ref> He also said that he had "never met his equal, and can compare him only with Euler or Jacobi".<ref name="cola1" /> Hardy thought Ramanujan worked in a 19th-century style, where arriving at correct formulas was more important than systematic formal theories. Hardy thought his achievements were greatest in algebra, especially hypergeometric series and continued fractions.<ref name=":0">{{Cite journal |last=Hardy |first=G. H. |date=March 1937 |title=The Indian Mathematician Ramanujan |url=https://www.tandfonline.com/doi/full/10.1080/00029890.1937.11987940 |journal=The American Mathematical Monthly |language=en |volume=44 |issue=3 |pages=137–155 |doi=10.1080/00029890.1937.11987940 |issn=0002-9890}}</ref>

{{blockquote|It is possible that the great days of formulas are finished, and that Ramanujan ought to have been born 100 years ago; but he was by far the greatest formalist of his time. There have been a good many more important, and I suppose one must say greater, mathematicians than Ramanujan during the last 50 years, but not one who could stand up to him on his own ground. Playing the game of which he knew the rules, he could give any mathematician in the world fifteen.<ref name=":0" />}}

He discovered fewer new things in analysis, possibly because he lacked the formal education and did not find books to learn it from, but rediscovered many results, including the [[prime number theorem]]. In analysis, he worked on the elliptic functions and the analytic theory of numbers. In [[analytic number theory]], he was as imaginative as usual, but much of what he imagined was wrong. Hardy blamed this on the inherent difficulty of analytic number theory, where imagination had led many great mathematicians astray. In analytic number theory, rigorous proof is more important than imagination, the opposite of Ramanujan's style. His "one great failure" is that he knew "nothing at all about the theory of [[analytic function]]s".<ref name=":0" />

Littlewood reportedly said that helping Ramanujan catch up with European mathematics beyond what was available in India was very difficult because each new point mentioned to Ramanujan caused him to produce original ideas that prevented Littlewood from continuing the lesson.<ref name="alyangar1995">{{Cite book |last=Alyangar |first=Srinivasa Ramanujan |url=https://books.google.com/books?id=Of5G0r6DQiEC&pg=PA51 |title=Ramanujan: Letters and Commentary |publisher=American Mathematical Society |year=1995 |isbn=0821891251 |editor-last=Berndt |editor-first=Bruce C. |series=History of Mathematics |volume=9 |page=51 |access-date=2023-02-22 |editor-last2=Rankin |editor-first2=Robert Alexander}}</ref>

K. Srinivasa Rao has said,<ref>{{cite web| url=https://www.imsc.res.in/~rao/ramanujan.html| author=Rao, K Srinivasa| title=Srinivasa Ramanujan (22 December 1887 – 26 April 1920)| url-status=dead| archive-url=https://web.archive.org/web/20120416085328/https://www.imsc.res.in/~rao/ramanujan.html| archive-date=16 April 2012| df=dmy-all| access-date=23 June 2018}}</ref> "As for his place in the world of Mathematics, we quote Bruce C. Berndt: '[[Paul Erdős]] has passed on to us Hardy's personal ratings of mathematicians. Suppose that we rate mathematicians on the basis of pure talent on a scale from 0 to 100. Hardy gave himself a score of 25, J. E. Littlewood 30, [[David Hilbert]] 80 and Ramanujan 100.{{'"}} During a May 2011 lecture at [[IIT Madras]], Berndt said that over the last 40 years, as nearly all of Ramanujan's conjectures had been proven, there had been greater appreciation of Ramanujan's work and brilliance, and that Ramanujan's work was now pervading many areas of modern mathematics and physics.<ref name="Fox"/><ref>{{cite web|via=YouTube|title=Bruce Berndt on "Ramanujan's Lost Notebook", IIT Madras, 24th May 2011|date=30 May 2011 |url= https://www.youtube.com/watch?v=5uCAuwOQoxA |url-status=live|archive-url= https://web.archive.org/web/20151206114320/https://www.youtube.com/watch?v=5uCAuwOQoxA |archive-date=6 December 2015}}</ref>

==Posthumous recognition==
{{further|List of things named after Srinivasa Ramanujan}}
[[File:Srinivasa Ramanujam bust BITM.JPG|thumb|upright|Bust of Ramanujan in the garden of [[Birla Industrial & Technological Museum]] in [[Kolkata]], India]]
The year after his death, ''[[Nature (journal)|Nature]]'' listed Ramanujan among other distinguished scientists and mathematicians on a "Calendar of Scientific Pioneers" who had achieved eminence.<ref>{{Cite journal | title =Calendar of Scientific Pioneers | journal =[[Nature (journal)|Nature]]| volume =107  | issue =2686| pages =252–254 | date =21 April 1921 | bibcode =1921Natur.107..252.| doi =10.1038/107252b0| doi-access =free}}</ref> Ramanujan's home state of [[Tamil Nadu]] celebrates 22 December (Ramanujan's birthday) as 'State IT Day'. Stamps picturing Ramanujan were issued by the [[government of India]] in 1962, 2011, 2012 and 2016.<ref>[https://commons.wikimedia.org/wiki/Category:Srinivasa_Ramanujan_on_stamps Srinivasa Ramanujan on stamps]. commons.wikimedia.org</ref>

Since Ramanujan's centennial year, his birthday, 22 December, has been annually celebrated as Ramanujan Day by the [[Government Arts College, Kumbakonam]], where he studied, and at the [[Indian Institute of Technology Madras|IIT Madras]] in [[Chennai]]. The [[International Centre for Theoretical Physics]] (ICTP) has created a prize in Ramanujan's name for young mathematicians from developing countries in cooperation with the [[International Mathematical Union]], which nominates members of the prize committee. [[Shanmugha Arts, Science, Technology & Research Academy|SASTRA University]], a private university based in [[Tamil Nadu]], has instituted the [[SASTRA Ramanujan Prize]] of [[US$]]10,000 to be given annually to a mathematician not exceeding age 32 for outstanding contributions in an area of mathematics influenced by Ramanujan.<ref name=src/>

Based on the recommendations of a committee appointed by the University Grants Commission (UGC), Government of India, the Srinivasa Ramanujan Centre, established by SASTRA, has been declared an off-campus centre under the ambit of SASTRA University. House of Ramanujan Mathematics, a museum of Ramanujan's life and work, is also on this campus. SASTRA purchased and renovated the house where Ramanujan lived at Kumabakonam.<ref name=src>{{cite web|url=https://src.sastra.edu/index.php?option=com_content&view=article&id=234&Itemid=345|title=Sastra University – Srinivasa Ramanujan Center – About Us|url-status=dead|archive-url=https://web.archive.org/web/20170615164913/https://src.sastra.edu/index.php?option=com_content&view=article&id=234&Itemid=345|archive-date=15 June 2017|access-date=23 June 2018}}</ref>

In 2011, on the 125th anniversary of his birth, the Indian government declared that 22 December will be celebrated every year as ''National Mathematics Day''.<ref>{{cite web|url=https://ibnlive.in.com/generalnewsfeed/news/singhs-visit-first-to-the-state-after-the-congressdmk/942941.html|archive-url=https://archive.today/20120715010820/http://ibnlive.in.com/generalnewsfeed/news/singhs-visit-first-to-the-state-after-the-congressdmk/942941.html|url-status=dead|archive-date=15 July 2012|title=Singh's first visit to the state|work=CNN IBN|location=India|access-date=12 April 2016|date=26 December 2011}}</ref> Then Indian Prime Minister [[Manmohan Singh]] also declared that 2012 would be celebrated as [[National Mathematics Year]] and 22 December as [[National Mathematics Day (India)|National Mathematics Day]] of India.<ref>{{cite web|url=https://gauravtiwari.org/2011/12/28/2012/|title=Welcome 2012 – The National Mathematical Year in India|location=India|url-status=dead|archive-url=https://web.archive.org/web/20171206214920/https://gauravtiwari.org/2011/12/28/2012/|archive-date=6 December 2017|date=28 December 2011|access-date=6 December 2017}}</ref>

[[Ramanujan IT City]] is an information technology (IT) [[special economic zone]] (SEZ) in [[Chennai]] that was built in 2011. Situated next to the [[TIDEL Park|Tidel Park]], it includes {{convert|25|acre}} with two zones, with a total area of {{convert|5.7|e6sqft|sqm}}, including {{convert|4.5|e6sqft|sqm}} of office space.<ref>{{Cite web|url=https://property.jll.co.in/office-lease/chennai/perungudi/ramanujan-it-city-hardy-tower-ind-p-000f4f|title=Ramanujan IT City - Hardy Tower|website=JLL Property India}}</ref>

==Commemorative postal stamps==
Commemorative stamps released by [[India Post]] (by year):
{{multiple image
 | align = center
 | direction = horizontal
 | header =

 | image1 = Srinivasa Ramanujan 1962 stamp of India.jpg
 | width1 = 200
 | alt1 = 
 | caption1 = 1962

 | image2 =Srinivasa Ramanujan 2011 stamp of India.jpg
 | width2 = 220
 | alt2 = 
 | caption2 = 2011

 | image3 =Srinivasa Ramanujan 2012 stamp of India.jpg
 | width3 = 270
 | alt3 = 
 | caption3 = 2012

 | image4 =Srinivasa Ramanujan 2016 stamp of India.jpg
 | width4 = 150
 | alt4 = 
 | caption4 = 2016

}}

==In popular culture==
* ''The Man Who Loved Numbers'' is a 1988 PBS NOVA documentary about Ramanujan (S15, E9).<ref>{{Cite web |url=https://www.imdb.com/title/tt1447049/ |title=The Man Who Loved Numbers |date=22 March 1988 |website=IMDB |access-date=24 September 2021}}</ref>
* ''[[The Man Who Knew Infinity]]'' is a 2015 film based on [[Robert Kanigel|Kanigel's]] [[The Man Who Knew Infinity (book)|book of the same name]]. British actor [[Dev Patel]] portrays Ramanujan.<ref>{{cite news|title=Cannes: Dev Patel to Star as Famed Indian Mathematician |url=https://www.hollywoodreporter.com/news/cannes-dev-patel-star-as-524307 |publisher=hollywoodreporter.com |url-status=live|archive-url=https://web.archive.org/web/20140109091247/https://www.hollywoodreporter.com/news/cannes-dev-patel-star-as-524307 |archive-date=9 January 2014}}</ref><ref>{{cite magazine |title=Jeremy Irons to Co-star in 'The Man Who Knew Infinity' |url=https://variety.com/2013/film/news/jeremy-irons-to-co-star-in-the-man-who-knew-infinity-1200921417/ |magazine=variety.com |url-status=live |archive-url=https://web.archive.org/web/20171012150848/https://variety.com/2013/film/news/jeremy-irons-to-co-star-in-the-man-who-knew-infinity-1200921417/ |archive-date=12 October 2017 |author=Barraclough, Leo |date=5 December 2013}}</ref><ref>{{cite magazine |title=Dev Patel's 'The Man Who Knew Infinity' Moves to Production After 8 Years in Development |url=https://variety.com/2014/film/news/dev-patels-the-man-who-knew-infinity-moves-to-production-after-8-years-in-development-1201262436/ |magazine=variety.com |url-status=live |archive-url=https://web.archive.org/web/20170704075230/https://variety.com/2014/film/news/dev-patels-the-man-who-knew-infinity-moves-to-production-after-8-years-in-development-1201262436/ |archive-date=4 July 2017 |author=McNary, Dave |date=15 July 2014}}</ref>
* ''[[Ramanujan (film)|Ramanujan]]'', an Indo-British collaboration film chronicling Ramanujan's life, was released in 2014 by the independent film company [[Ramanujan (film)#Camphor Cinema|Camphor Cinema]].<ref>{{cite news |url=https://www.indiatimes.com/entertainment/regional/ramanujan-makers-shoot-in-his-house-87409.html |title='Ramanujan' Makers Shoot in His House |work=India Times |access-date=12 July 2013 |url-status=live |archive-url=https://web.archive.org/web/20130711135855/https://www.indiatimes.com/entertainment/regional/ramanujan-makers-shoot-in-his-house-87409.html |archive-date=11 July 2013}}</ref> The cast and crew include director [[Gnana Rajasekaran]], cinematographer [[Sunny Joseph]] and editor [[B. Lenin]].<ref>{{cite web |url=https://www.boxofficeindia.co.in/camphor-cinema-presents-their-first-film-ramanujan/ |title=Camphor Cinema Presents Their First Film Ramanujan |publisher=Select Publishing Company |work=Box Office India |access-date=12 July 2013 |url-status=dead |archive-url=https://web.archive.org/web/20130820212249/https://www.boxofficeindia.co.in/camphor-cinema-presents-their-first-film-ramanujan/ |archive-date=20 August 2013 |date=11 June 2013}}</ref><ref>{{cite web |url=https://zeenews.india.com/entertainment/regional/makers-of-ramanujan-shoot-in-genius-house_138461.htm |title=Makers of 'Ramanujan' shoot in genius' house |publisher=Zee Media Corporation Ltd |work=Z News |access-date=12 July 2013 |url-status=dead |archive-url=https://web.archive.org/web/20130708015845/https://zeenews.india.com/entertainment/regional/makers-of-ramanujan-shoot-in-genius-house_138461.htm |archive-date=8 July 2013}}</ref> Indian and English stars [[Abhinay Vaddi]], [[Suhasini Maniratnam]], [[Bhama Kurup|Bhama]], Kevin McGowan and [[Michael Lieber]] star in pivotal roles.<ref>{{cite news |url=https://www.thehindu.com/features/friday-review/travails-of-a-genius/article4857108.ece |title=Travails of a genius |last=Krishnamachari |first=Suganthy |date=27 June 2013 |work=The Hindu |location=Chennai, India |access-date=12 July 2013 |url-status=live |archive-url=https://web.archive.org/web/20130701114803/https://www.thehindu.com/features/friday-review/travails-of-a-genius/article4857108.ece |archive-date=1 July 2013}}</ref>
* Nandan Kudhyadi directed the Indian documentary films ''The Genius of Srinivasa Ramanujan'' (2013) and ''Srinivasa Ramanujan: The Mathematician and His Legacy'' (2016) about the mathematician.<ref>{{cite web |title=Pune-based filmmaker wins 3 awards at National Science Film festival |url=https://indianexpress.com/article/cities/pune/pune-based-filmmaker-wins-3-awards-at-national-science-film-festival-4545771/ |website=[[The Indian Express]] |language=en |date=27 February 2017}}</ref> 
* ''Ramanujan (The Man Who Reshaped 20th Century Mathematics)'', an Indian docudrama film directed by Akashdeep released in 2018.<ref>{{cite web |title=Ramanujan (The Man who reshaped 20th Century Mathematics) (2018) |url=https://indiancine.ma/CGOD/info |website=Indiancine.ma}}</ref>
* M. N. Krish's thriller novel ''The Steradian Trail'' weaves Ramanujan and his accidental discovery into its plot connecting religion, mathematics, finance and economics.<ref>{{cite news |title=Racy read |url=https://www.thehindu.com/books/literary-review/racy-read/article6667634.ece |date=7 December 2014 |access-date=30 April 2016 |newspaper=The Hindu |last1=Basu |first1=Kankana}}</ref><ref>{{cite web |title=Crime in a World of High Science |url=https://www.newindianexpress.com/cities/bengaluru/Crime-in-a-World-of-High-Science/2014/09/16/article2432584.ece |date=16 September 2014 |access-date=30 April 2016 |url-status=dead |archive-url=https://web.archive.org/web/20160415210040/https://www.newindianexpress.com/cities/bengaluru/Crime-in-a-World-of-High-Science/2014/09/16/article2432584.ece |archive-date=15 April 2016}}</ref>
* ''Partition'', a play by Ira Hauptman about Hardy and Ramanujan, was first performed in 2013.<ref>{{cite journal |author=Ribet, Kenneth A. |date=December 2003 |journal=Notices of the AMS |pages=1407–1408 |volume=50 |issue=1 |url=https://www.ams.org/notices/200311/rev-ribet.pdf |title=Theater Review. Partition |access-date=27 September 2016 |url-status=live |archive-url=https://web.archive.org/web/20161006222734/https://www.ams.org/notices/200311/rev-ribet.pdf |archive-date=6 October 2016}}</ref><ref>{{cite web |url=https://variety.com/2003/legit/reviews/partition-1200541626/ |title=Review: 'Partition' |first=Dennis |last=Harvey |date=18 May 2003 |access-date=23 March 2017 |url-status=live |archive-url=https://web.archive.org/web/20161006224120/https://variety.com/2003/legit/reviews/partition-1200541626/ |archive-date=6 October 2016}}</ref><ref>{{cite web |url=https://www.thehindu.com/2003/05/26/stories/2003052604331200.htm |title=Partitions – a play on Ramanujan |date=26 May 2003 |url-status=dead |archive-url=https://web.archive.org/web/20080720071318/https://www.thehindu.com/2003/05/26/stories/2003052604331200.htm |work=[[The Hindu]] |archive-date=20 July 2008}}</ref><ref>{{cite news |url=https://www.thehindu.com/features/metroplus/an-ode-to-a-genius/article6708231.ece |title=An ode to a genius |first=SRAVASTI |last=DATTA |access-date=23 March 2017 |newspaper=The Hindu |date=19 December 2014}}</ref>
* The play ''First Class Man'' by Alter Ego Productions<ref>{{cite web |url=https://www.alteregoproductions.org/blog/2006/06/alteregos_new_theater_season_b.htm |archive-url=https://web.archive.org/web/20070629012019/https://www.alteregoproductions.org/blog/2006/06/alteregos_new_theater_season_b.htm |url-status=usurped |archive-date=29 June 2007 |title=First Class Man |publisher=Alteregoproductions.org |access-date=20 November 2012 }}</ref> was based on David Freeman's ''First Class Man''. The play centres around Ramanujan and his complex and dysfunctional relationship with Hardy. On 16 October 2011 it was announced that [[Roger Spottiswoode]], best known for his [[James Bond film]] ''[[Tomorrow Never Dies]]'', is working on the film version, starring [[Siddharth (actor)|Siddharth]].<ref>{{cite news |url=https://www.thehindu.com/news/national/article2541084.ece |title=News / National: James Bond director to make film on Ramanujan |work=[[The Hindu]] |location=India |access-date=18 October 2011 |date=16 October 2011 |url-status=live |archive-url=https://web.archive.org/web/20111017180258/https://www.thehindu.com/news/national/article2541084.ece |archive-date=17 October 2011}}</ref>
* ''[[A Disappearing Number]]'' is a British stage production by the company [[Complicite]] that explores the relationship between Hardy and Ramanujan.<ref>{{Cite news |url=https://www.npr.org/templates/story/story.php?storyId=128513493 |title='Disappearing Number': A Vivid Theatrical Equation |last=Lunden |first=Jeff |date=15 July 2010 |work=Morning Edition |publisher=[[National Public Radio]] |access-date=24 April 2018 }}</ref>
* [[David Leavitt]]'s novel ''[[The Indian Clerk]]'' explores the events following Ramanujan's letter to Hardy.<ref>{{cite news |url=https://www.nytimes.com/2007/09/16/books/review/Freudenberger-t.html |title=Lust for Numbers |author=Freudenberger, Nell |work=The New York Times |date=16 September 2007 |access-date=4 September 2011 |url-status=live |archive-url=https://web.archive.org/web/20120110114721/https://www.nytimes.com/2007/09/16/books/review/Freudenberger-t.html |archive-date=10 January 2012}}</ref><ref>{{cite news |title=Adding up to a life |url=https://www.theguardian.com/books/2008/jan/26/fiction |author=Taylor, D. J. |work=The Guardian |location=UK |date=26 January 2008 |access-date=4 September 2011|url-status=live|archive-url=https://web.archive.org/web/20141006140353/https://www.theguardian.com/books/2008/jan/26/fiction1|archive-date=6 October 2014}}</ref>
* [[Google]] honoured Ramanujan on his 125th birth anniversary by replacing its logo with a [[Google Doodles|doodle]] on its home page.<ref>{{cite news |title=Google doodles for Ramanujan's 125th birthday |url=http://timesofindia.indiatimes.com/tech/tech-news/internet/Google-doodles-for-Ramanujans-125th-birthday/articleshow/17716898.cms |access-date=22 December 2012 |newspaper=Times of India |date=22 December 2012 |archive-url=https://web.archive.org/web/20121222073142/http://timesofindia.indiatimes.com/tech/tech-news/internet/Google-doodles-for-Ramanujans-125th-birthday/articleshow/17716898.cms |url-status=dead |archive-date=22 December 2012}}</ref><ref>{{Cite web |url=https://doodles.google/doodle/srinivasa-ramanujans-125th-birthday/ |title=Srinivasa Ramanujan's 125th Birthday |website=www.google.com |access-date=30 April 2016 |url-status=live |archive-url=https://web.archive.org/web/20160510194306/https://www.google.com/doodles/srinivasa-ramanujans-125th-birthday |archive-date=10 May 2016}}</ref>
* Ramanujan was mentioned in the 1997 film ''[[Good Will Hunting]]'', in a scene where professor Gerald Lambeau ([[Stellan Skarsgård]]) explains to Sean Maguire ([[Robin Williams]]) the genius of Will Hunting ([[Matt Damon]]) by comparing him to Ramanujan.<ref>{{Cite web |url=https://www.psychologytoday.com/us/blog/psychology-masala/201802/legendary-creative-math-genius-srinivasa-ramanujan |title=A Legendary Creative Math Genius: Srinivasa Ramanujan |last=Kumar |first=V. Krishna |date=2 February 2018 |website=Psychology Today |access-date=24 April 2018}}</ref>

==Selected papers==

{{Div col|colwidth=35em}}

*{{Cite journal|last=Ramanujan|first=S. |year=1914|title=Some definite integrals|journal=Messenger Math.|volume=44 |pages=10–18|url=https://archive.org/details/messengerofmathe44cambuoft/page/10/mode/2up|doi=|s2cid=}}
*{{Cite journal|last=Ramanujan|first=S. |year=1914|title=Some definite integrals connected with Gauss's sums|journal=Messenger Math.|volume=44 |pages=75–85|url=https://archive.org/details/messengerofmathe44cambuoft/page/10/mode/2up}}
*{{Cite journal|last=Ramanujan|first=S. |year=1915|title=On certain infinite series|journal=Messenger Math.|volume=45|pages=11–15|url=https://archive.org/details/messengerofmathe45cambuoft/page/10/mode/2up|doi=|s2cid=}}
*{{Cite journal|last1=Ramanujan|first1=S.|author-link1=<!--Srinivasa Ramanujan-->|year=1915|title=Highly Composite Numbers|journal=Proceedings of the London Mathematical Society |volume=14|issue=1|pages =347–409|doi=10.1112/plms/s2_14.1.347|url=https://zenodo.org/record/1433496 |doi-access=}}
*{{Cite journal|last=Ramanujan|first=S. |year=1915|title=On the number of divisors of a number|journal=The Journal of the Indian Mathematical Society |volume=7|issue=4 |pages=131–133|url=https://archive.org/details/dli.ernet.504289/page/131/mode/2up|doi=|s2cid=}}
*{{Cite journal|last=Ramanujan|first=S. |year=1915|title=Short Note: On the sum of the square roots of the first ''n'' natural numbers|journal=The Journal of the Indian Mathematical Society |volume=7|issue=5 |pages=173–175|url=https://archive.org/details/dli.ernet.504289/page/173/mode/2up|doi=|s2cid=}}
*{{Cite journal|last=Ramanujan|first=S. |year=1916|title=Some formulae in the analytical theory of numbers|journal=Messenger Math.|volume=45|pages=81–84|url=https://archive.org/details/messengerofmathe45cambuoft/page/80/mode/2up|doi=|s2cid=}}
*{{Cite journal|last=Ramanujan|first=S. |year=1916|title=A Series for Euler's Constant γ|journal=Messenger Math.|volume=46|pages=73–80|url=https://archive.org/details/messengerofmathe46cambuoft/page/72/mode/2up|doi=|s2cid=}}
*{{Cite journal|last=Ramanujan|first=S. |year=1917|title=On the expression of numbers in the form {{math|1=ax<sup>2</sup> +  by<sup>2</sup> + cz<sup>2</sup> + du<sup>2</sup>}}|journal=[[Mathematical Proceedings of the Cambridge Philosophical Society]] |volume=19|pages=11–21|url=https://archive.org/details/proceedingsofcam1920191721camb/page/n23/mode/2up|doi=|s2cid=}}
*{{Cite journal|last1=Hardy|first1=G. H.|author-link1=<!--G. H. Hardy-->|last2=Ramanujan|first2=S.|author-link2=<!--Srinivasa Ramanujan-->|year=1917|title=Asymptotic Formulae for the Distribution of Integers of Various Types|journal=Proceedings of the London Mathematical Society |volume=16|issue=1|pages=112–132|doi=10.1112/plms/s2-16.1.112|url=https://zenodo.org/record/1553638 |doi-access=}}
*{{Cite journal|last1=Hardy|first1=G. H. |author-link1=G. H. Hardy|last2=Ramanujan|first2=Srinivasa |year=1918|title=Asymptotic Formulae in Combinatory Analysis|journal=Proceedings of the London Mathematical Society |volume=17|issue=1|pages=75–115|doi=10.1112/plms/s2-17.1.75 |url=https://zenodo.org/record/1560148 }}
*{{Cite journal|last1=Hardy|first1=G. H. |last2=Ramanujan|first2=Srinivasa |year=1918|title=On the coefficients in the expansions of certain modular functions|journal=Proc. R. Soc. A |volume=95|issue=667|pages=144–155|doi=10.1098/rspa.1918.0056|bibcode=1918RSPSA..95..144H |doi-access=free}}
*{{Cite journal|last=Ramanujan|first=Srinivasa |year=1919 |title=Some definite integrals|journal=The Journal of the Indian Mathematical Society |volume=11|issue=2 |pages=81–88|url=https://archive.org/details/in.ernet.dli.2015.509608/page/n89/mode/2up }}
*{{Cite journal|last=Ramanujan|first=S. |year=1919|title=A proof of Bertrand's postulate|journal=The Journal of the Indian Mathematical Society |volume=11|issue=5 |pages=181–183|url=https://archive.org/details/in.ernet.dli.2015.509608/page/n195/mode/2up }}
*{{Cite journal|last=Ramanujan|first=S. |year=1920|title=A class of definite integrals|journal=Quart. J. Pure. Appl. Math. |volume=48|issue= |pages=294–309|hdl=2027/uc1.$b417568?urlappend=%3Bseq=306 |url=https://hdl.handle.net/2027/uc1.$b417568?urlappend=%3Bseq=306%3Bownerid=9007199275659670-310 }}
*{{Cite journal|last=Ramanujan|first=S. |year=1921|title=Congruence properties of partitions|journal=Math. Z. |volume=9|issue= 1–2|pages=147–153|url=https://zenodo.org/record/1447425|doi=10.1007/BF01378341|s2cid=121753215}} <small>Posthumously published extract of a longer, unpublished manuscript.</small>
	
{{colend}}

==Further works of Ramanujan's mathematics==
* [[George E. Andrews]] and [[Bruce C. Berndt]], ''Ramanujan's Lost Notebook: Part I''  (Springer, 2005, {{ISBN|0-387-25529-X}})<ref>{{cite journal|author=Bressoud, David|author-link=David Bressoud|title=Review: ''Ramanujan's Lost Notebook, Part I'', by George Andrews and Bruce C. Berndt|journal=[[Bulletin of the American Mathematical Society]] |series=New Series|year=2006|volume=43|issue=4|pages=585–591|url=http://www.ams.org/journals/bull/2006-43-04/S0273-0979-06-01110-4/S0273-0979-06-01110-4.pdf |archive-url=https://web.archive.org/web/20150623055212/http://www.ams.org/journals/bull/2006-43-04/S0273-0979-06-01110-4/S0273-0979-06-01110-4.pdf |archive-date=2015-06-23 |url-status=live|doi=10.1090/s0273-0979-06-01110-4|doi-access=free}}</ref>
* George E. Andrews and Bruce C. Berndt, ''Ramanujan's Lost Notebook: Part II'', (Springer, 2008, {{ISBN|978-0-387-77765-8}})
* George E. Andrews and Bruce C. Berndt, ''Ramanujan's Lost Notebook: Part III'', (Springer, 2012, {{ISBN|978-1-4614-3809-0}})
* George E. Andrews and Bruce C. Berndt, ''Ramanujan's Lost Notebook: Part IV'', (Springer, 2013, {{ISBN|978-1-4614-4080-2}})
* George E. Andrews and Bruce C. Berndt, ''Ramanujan's Lost Notebook: Part V'', (Springer, 2018, {{ISBN|978-3-319-77832-7}})
* M. P. Chaudhary, ''A simple solution of some integrals given by Srinivasa Ramanujan,'' (Resonance: J. Sci. Education – publication of Indian Academy of Science, 2008)<ref>{{cite journal|title=A simple solution of some integrals given by Srinivasa Ramanujan |journal=Resonance|volume=13|issue=9|pages=882–884|url=http://www.ias.ac.in/listing/issues/reso.pdf |archive-url=https://web.archive.org/web/20181006000719/http://www.ias.ac.in/listing/issues/reso.pdf |archive-date=2018-10-06 |url-status=live}}</ref>
* M.P. Chaudhary, ''Mock theta functions to mock theta conjectures,'' SCIENTIA, Series A: Math. Sci., (22)(2012) 33–46.
* M.P. Chaudhary, ''On modular relations for the Roger-Ramanujan type identities,'' Pacific J. Appl. Math., 7(3)(2016) 177–184.

==Selected publications on Ramanujan and his work==
{{refbegin|30em}}
* {{Cite book |last=Berndt |first=Bruce C. |editor-last1=Butzer |editor-first1=P. L. |editor-last2=Oberschelp |editor-first2=W. |editor-last3=Jongen |editor-first3=H. Th. |year=1998 |title=Charlemagne and His Heritage: 1200 Years of Civilization and Science in Europe |publisher=Brepols Verlag |location=Turnhout, Belgium |pages=119–146 |isbn=978-2-503-50673-9 |url=https://www.math.uiuc.edu/~berndt/articles/aachen.pdf |archive-url=https://web.archive.org/web/20040909045617/http://www.math.uiuc.edu/~berndt/articles/aachen.pdf |archive-date=2004-09-09 |url-status=live }}
* {{Cite book |last1=Berndt |first1=Bruce C. |last2=Rankin |first2=Robert A. |year=1995 |title=Ramanujan: Letters and Commentary |volume=9 |publisher=[[American Mathematical Society]] |location=Providence, Rhode Island |isbn=978-0-8218-0287-8 }}
* {{Cite book |last1=Berndt |first1=Bruce C. |first2=Robert A. |last2=Rankin |author2-link=Robert Alexander Rankin |title=Ramanujan: Essays and Surveys |volume=22 |location=Providence, Rhode Island |publisher=[[American Mathematical Society]] |year=2001 |isbn=978-0-8218-2624-9 |author-link=Bruce C. Berndt |url-access=registration |url=https://archive.org/details/trent_0116405145958 }}
* {{Cite book |last=Berndt |first=Bruce C. |year=2006 |title=Number Theory in the Spirit of Ramanujan |volume=9 |publisher=[[American Mathematical Society]] |location=Providence, Rhode Island |isbn=978-0-8218-4178-5 }}
* {{Cite book |last=Berndt |first=Bruce C. |year=1985 |title=Ramanujan's Notebooks: Part I |publisher=Springer |location=New York |isbn=978-0-387-96110-1 }}
* {{Cite book |last=Berndt |first=Bruce C. |year=1999 |title=Ramanujan's Notebooks: Part II |publisher=Springer |location=New York |isbn=978-0-387-96794-3 }}
* {{Cite book |last=Berndt |first=Bruce C. |year=2004 |title=Ramanujan's Notebooks: Part III |publisher=Springer |location=New York |isbn=978-0-387-97503-0 |url-access=registration |url=https://archive.org/details/ramanujansnotebo0000rama }}
* {{Cite book |last=Berndt |first=Bruce C. |year=1993 |title=Ramanujan's Notebooks: Part IV |publisher=Springer |location=New York |isbn=978-0-387-94109-7 }}
* {{Cite book |last=Berndt |first=Bruce C. |year=2005 |title=Ramanujan's Notebooks: Part V |publisher=Springer |location=New York |isbn=978-0-387-94941-3 }}
* {{Cite journal|last=Hardy|first=G. H.|date=March 1937|title=The Indian Mathematician Ramanujan|journal=The American Mathematical Monthly |volume=44|issue=3|pages=137–155|jstor=2301659|doi=10.2307/2301659}}
* {{Cite book |last=Hardy |first=G. H. |year=1978 |title=Ramanujan |publisher=Chelsea Pub. Co. |location=New York |isbn=978-0-8284-0136-4 }}
* {{Cite book |last=Hardy |first=G. H. |year=1999 |title=Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work |publisher=American Mathematical Society |location=Providence, Rhode Island |isbn=978-0-8218-2023-0 }}
* {{Cite book |last=Henderson |first=Harry |year=1995 |title=Modern Mathematicians |publisher=Facts on File Inc. |location=New York |isbn=978-0-8160-3235-8 |url-access=registration |url=https://archive.org/details/modernmathematic00hend }}
* {{Cite book |last=Kanigel |first=Robert |author-link=Robert Kanigel|year=1991 |title=[[The Man Who Knew Infinity: a Life of the Genius Ramanujan]] |location=New York |publisher=Charles Scribner's Sons |isbn=978-0-684-19259-8 }}
* {{Cite book |last=Leavitt |first=David |author-link=David Leavitt |year=2007 |title=The Indian Clerk |edition=paperback |publisher=Bloomsbury |location=London |isbn=978-0-7475-9370-6 |title-link=The Indian Clerk }}
* {{Cite book |last=Narlikar |first=Jayant V. |author-link=Jayant V. Narlikar |year=2003 |title=Scientific Edge: the Indian Scientist From Vedic to Modern Times |publisher=Penguin Books |location=New Delhi, India |isbn=978-0-14-303028-7 }}
* {{cite book|last1= Ono |first1= Ken |author-link1= Ken Ono |last2= Aczel|first2= Amir D. |author-link2= Amir Aczel |title= My Search for Ramanujan: How I Learned to Count |date= 13 April 2016 |publisher= [[Springer Science+Business Media|Springer]] |isbn= 978-3319255668}}
* {{Cite report |last=Sankaran |first=T. M. |year=2005 |title=Srinivasa Ramanujan- Ganitha lokathile Mahaprathibha |publisher=[[Kerala Sasthra Sahithya Parishad]] |location=Kochi, India |language=ml }}
{{refend}}

==Selected publications on works of Ramanujan==
{{refbegin|30em}}
* {{Cite book|title=Collected Papers of Srinivasa Ramanujan |author-first1=Srinivasa |author-last1=Ramanujan |author-first2=G. H. |author-last2=Hardy |author-first3=P. V. |author-last3=Seshu Aiyar |author-first4=B. M. |author-last4=Wilson |author-link4=Bertram Martin Wilson |author-first5=Bruce C. |author-last5=Berndt |publisher=AMS |year=2000 |isbn=978-0-8218-2076-6}}
:This book was originally published in 1927<ref>{{cite journal|title=''Collected Papers of Srinivasa Ramanujan'' edited by G. H. Hardy, P. V. Seshu Aiyar and B. M. Wilson |author=Bell, E. T.|author-link=Eric Temple Bell|journal=Bulletin of the American Mathematical Society |volume=34|issue=6|year=1928|pages=783–784|doi=10.1090/S0002-9904-1928-04651-7|doi-access=free}}</ref> after Ramanujan's death. It contains the 37 papers published in professional journals by Ramanujan during his lifetime. The third reprint contains additional commentary by Bruce C. Berndt.
* {{Cite book|title=Notebooks (2 Volumes) |author=S. Ramanujan |publisher=Tata Institute of Fundamental Research |place=Bombay |year=1957}}
:These books contain photocopies of the original notebooks as written by Ramanujan.
* {{Cite book|title=The Lost Notebook and Other Unpublished Papers |author=S. Ramanujan |publisher=Narosa |place=New Delhi |year=1988|isbn=978-3-540-18726-4}}
:This book contains photocopies of the pages of the "Lost Notebook".
* [https://www.imsc.res.in/~rao/ramanujan/collectedpapers/question/qJIMS.htm Problems posed by Ramanujan], Journal of the Indian Mathematical Society.
* {{Cite book|title=Notebooks (2 Volumes) |author=S. Ramanujan |publisher=Tata Institute of Fundamental Research |place=Bombay |year=2012}}
:This was produced from scanned and microfilmed images of the original manuscripts by expert archivists of Roja Muthiah Research Library, Chennai.
{{refend}}

==See also==
{{Portal|Mathematics|Biography|India}}
{{Div col}}
* [[1729 (number)]]
* [[Brown numbers]]
* [[List of amateur mathematicians]]
* [[List of Indian mathematicians]]
* [[Ramanujan graph]]
* [[Ramanujan summation]]
* [[Ramanujan's constant]]
* [[Ramanujan's ternary quadratic form]]
* [[Rank of a partition]]
{{Div col end}}

==Footnotes==
{{notelist}}

==References==
{{Reflist}}

==External links==
{{Sister project links| wikt = no| b = no| q = Srinivasa Ramanujan | s = Author:Srinivasa Ramanujan | commons = Srinivasa Ramanujan | n = no| v = no| species = no| display = Srinivasa Ramanujan | author = no}}

===Media links===
* {{Cite news |title=Film to celebrate mathematics genius |work=BBC |url=https://news.bbc.co.uk/2/hi/south_asia/4811920.stm |access-date=24 August 2006 |date=16 March 2006 |first=Soutik |last=Biswas}}
* [https://devbenegal.com/2006/03/15/feature-film-on-math-genius-ramanujan/ Feature Film on Mathematics Genius Ramanujan by Dev Benegal and Stephen Fry]
* [https://www.bbc.co.uk/radio4/science/further5.shtml BBC radio programme about Ramanujan – episode 5]
* [https://archive.org/details/Ramanujan A biographical song about Ramanujan's life]
* {{cite web |url=https://www.youtube.com/watch?v=ixra6x1XdjE |title=Why Did This Mathematician's Equations Make Everyone So Angry? |author= |date=11 April 2022 |via=YouTube |publisher=Thoughty2 |access-date=29 June 2022 |quote=}}

===Biographical links===
* {{MathGenealogy|id=91561}}
* {{MacTutor | id=Ramanujan}}
* {{ScienceWorldBiography | urlname=Ramanujan | title=Ramanujan, Srinivasa (1887–1920)}}
* [https://www.usna.edu/Users/math/meh/ramanujan.html A short biography of Ramanujan]
* [https://web.archive.org/web/20180805161945/http://srinivasaramanujan.in/ "Our Devoted Site for Great Mathematical Genius"]

===Other links===
* {{cite journal|url=https://writings.stephenwolfram.com/2016/04/who-was-ramanujan/|title= Who Was Ramanujan?|first=Stephen|last=Wolfram|journal= Stephen Wolfram Writings|author-link=Stephen Wolfram|date=27 April 2016}}
* A Study Group For Mathematics: [https://archive.today/20121204163504/http://groups.yahoo.com/group/srinivasaramanujan/ Srinivasa Ramanujan Iyengar]
* [https://web.archive.org/web/20020126193110/https://www.math.ufl.edu/~frank/ramanujan.html ''The Ramanujan Journal''] – An international journal devoted to Ramanujan
* [https://web.archive.org/web/20051125142618/https://www.mathunion.org/General/Prizes/ International Math Union Prizes], including a Ramanujan Prize
* Hindu.com: [https://web.archive.org/web/20050120000345/http://www.hindu.com/mag/2004/12/26/stories/2004122600610400.htm Norwegian and Indian mathematical geniuses], [https://www.hindu.com/thehindu/br/2003/08/26/stories/2003082600120300.htm Ramanujan – Essays and Surveys] {{Webarchive|url=https://web.archive.org/web/20121106022700/http://www.hindu.com/thehindu/br/2003/08/26/stories/2003082600120300.htm |date=6 November 2012 }}, [https://web.archive.org/web/20040103074953/http://www.hindu.com/2003/12/22/stories/2003122204061100.htm Ramanujan's growing influence], [https://web.archive.org/web/20040628115315/http://www.hindu.com/thehindu/mag/2002/12/22/stories/2002122200040400.htm Ramanujan's mentor]
* Hindu.com: [https://www.thehindu.com/life-and-style/metroplus/article884010.ece The sponsor of Ramanujan]
* {{Cite journal
 |author1=Bruce C. Berndt |author2=Robert A. Rankin
 | title = The Books Studied by Ramanujan in India
 | journal = [[American Mathematical Monthly]]
 | volume = 107
 | year = 2000
 | issue = 7
 | pages = 595–601
 | doi = 10.2307/2589114
 | mr = 1786233
 | jstor = 2589114
 }}
* [https://www.maa.org/news/030807puzzlesolved.html "Ramanujan's mock theta function puzzle solved"]
* [https://www.imsc.res.in/~rao/ramanujan/contentindex.html Ramanujan's papers and notebooks]
* [https://web.archive.org/web/20171211191400/http://www.cecm.sfu.ca/organics/papers/borwein/paper/html/local/ramnotebook.html Sample page from the second notebook]
* [https://web.archive.org/web/20120319203227/http://www.friedeye.com/2011/01/01/ramanujan/ Ramanujan] on ''Fried Eye''
* {{cite web|last=Clark|first=Alex|title=163 and Ramanujan Constant|url=https://www.numberphile.com/videos/163.html|work=Numberphile|publisher=[[Brady Haran]]|access-date=23 June 2018|archive-url=https://web.archive.org/web/20180204091812/http://www.numberphile.com/videos/163.html|archive-date=4 February 2018|url-status=dead}}

{{Indian mathematics}}
{{Authority control}}

{{DEFAULTSORT:Srinivasa, Ramanujan}}
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