Srinivasa Ramanujan

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id="siteNotice"></div> </div> <div class="pre-content heading-holder"> <div class="page-heading"> <h1 id="firstHeading" class="firstHeading mw-first-heading">Srinivasa Ramanujan</h1> <div class="tagline"></div> </div> <ul id="p-associated-pages" class="minerva__tab-container"> <li class="minerva__tab selected"><a class="minerva__tab-text" href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" rel="" data-event-name="tabs.subject">Article</a></li> <li class="minerva__tab "><a class="minerva__tab-text" href="https://en-m-wikipedia-org.translate.goog/wiki/Talk:Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" rel="discussion" data-event-name="tabs.talk">Talk</a></li> </ul> <nav class="page-actions-menu"> <ul id="p-views" class="page-actions-menu__list"> <li id="language-selector" class="page-actions-menu__list-item"><a role="button" 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div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style> <div role="note" class="hatnote navigation-not-searchable"> "Ramanujan" redirects here. For other uses, see <a href="https://en-m-wikipedia-org.translate.goog/wiki/Ramanujan_(disambiguation)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-disambig" title="Ramanujan (disambiguation)">Ramanujan (disambiguation)</a>. </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"> <div role="note" class="hatnote navigation-not-searchable"> In this <a href="https://en-m-wikipedia-org.translate.goog/wiki/Indian_name?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Indian name">Indian name</a>, the name Srinivasa is a <a href="https://en-m-wikipedia-org.translate.goog/wiki/Patronymic?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Patronymic">patronymic</a>, and the person should be referred to by the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Given_name?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Given name">given name</a>, Ramanujan. </div> <style data-mw-deduplicate="TemplateStyles:r1257001546">.mw-parser-output .infobox-subbox{padding:0;border:none;margin:-3px;width:auto;min-width:100%;font-size:100%;clear:none;float:none;background-color:transparent}.mw-parser-output .infobox-3cols-child{margin:auto}.mw-parser-output .infobox .navbar{font-size:100%}@media screen{html.skin-theme-clientpref-night .mw-parser-output .infobox-full-data:not(.notheme)>div:not(.notheme)[style]{background:#1f1f23!important;color:#f8f9fa}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .infobox-full-data:not(.notheme) div:not(.notheme){background:#1f1f23!important;color:#f8f9fa}}@media(min-width:640px){body.skin--responsive .mw-parser-output .infobox-table{display:table!important}body.skin--responsive .mw-parser-output .infobox-table>caption{display:table-caption!important}body.skin--responsive .mw-parser-output .infobox-table>tbody{display:table-row-group}body.skin--responsive .mw-parser-output .infobox-table tr{display:table-row!important}body.skin--responsive .mw-parser-output .infobox-table th,body.skin--responsive .mw-parser-output .infobox-table td{padding-left:inherit;padding-right:inherit}}</style> Srinivasa Ramanujan Aiyangar<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-4">[a]</a> (22 December 1887 – 26 April 1920) was an Indian <a href="https://en-m-wikipedia-org.translate.goog/wiki/Mathematician?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Mathematician">mathematician</a>. Often regarded as one of the greatest mathematicians of all time, though he had almost no formal training in <a href="https://en-m-wikipedia-org.translate.goog/wiki/Pure_mathematics?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Pure mathematics">pure mathematics</a>, he made substantial contributions to <a href="https://en-m-wikipedia-org.translate.goog/wiki/Mathematical_analysis?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Mathematical analysis">mathematical analysis</a>, <a href="https://en-m-wikipedia-org.translate.goog/wiki/Number_theory?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Number theory">number theory</a>, <a href="https://en-m-wikipedia-org.translate.goog/wiki/Infinite_series?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Infinite series">infinite series</a>, and <a href="https://en-m-wikipedia-org.translate.goog/wiki/Continued_fraction?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Continued fraction">continued fractions</a>, including solutions to mathematical problems then considered unsolvable. <table class="infobox biography vcard"> <tbody> <tr> <th colspan="2" class="infobox-above" style="font-size:125%;"> <div class="fn"> Srinivasa Ramanujan </div> <div class="honorific-suffix" style="font-size: 77%; font-weight: normal;"> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Fellow_of_the_Royal_Society?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Fellow of the Royal Society">FRS</a> </div></th> </tr> <tr> <td colspan="2" class="infobox-image"><a href="https://en-m-wikipedia-org.translate.goog/wiki/File:Srinivasa_Ramanujan-Add._MS_a94_version2_(cropped).jpg?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d0/Srinivasa_Ramanujan-Add._MS_a94_version2_%28cropped%29.jpg/220px-Srinivasa_Ramanujan-Add._MS_a94_version2_%28cropped%29.jpg" decoding="async" width="220" height="293" class="mw-file-element" srcset="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://upload.wikimedia.org/wikipedia/commons/thumb/d/d0/Srinivasa_Ramanujan-Add._MS_a94_version2_%2528cropped%2529.jpg/330px-Srinivasa_Ramanujan-Add._MS_a94_version2_%2528cropped%2529.jpg 1.5x,https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://upload.wikimedia.org/wikipedia/commons/thumb/d/d0/Srinivasa_Ramanujan-Add._MS_a94_version2_%2528cropped%2529.jpg/440px-Srinivasa_Ramanujan-Add._MS_a94_version2_%2528cropped%2529.jpg 2x" data-file-width="2583" data-file-height="3445"></a> <div class="infobox-caption"> Ramanujan in 1913 </div></td> </tr> <tr> <th scope="row" class="infobox-label">Born</th> <td class="infobox-data"> <div style="display:inline" class="nickname"> Srinivasa Ramanujan Aiyangar </div> (1887-12-22)22 December 1887 <div style="display:inline" class="birthplace"> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Erode?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Erode">Erode</a>, <a href="https://en-m-wikipedia-org.translate.goog/wiki/Mysore_Princely_State?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Mysore Princely State">Mysore State</a>, <a href="https://en-m-wikipedia-org.translate.goog/wiki/British_Raj?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="British Raj">British India</a> (now in <a href="https://en-m-wikipedia-org.translate.goog/wiki/Tamil_Nadu?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Tamil Nadu">Tamil Nadu</a>, <a href="https://en-m-wikipedia-org.translate.goog/wiki/India?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="India">India</a>) </div></td> </tr> <tr> <th scope="row" class="infobox-label">Died</th> <td class="infobox-data">26 April 1920(1920-04-26) (aged 32) <div style="display:inline" class="deathplace"> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Kumbakonam?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Kumbakonam">Kumbakonam</a>, <a href="https://en-m-wikipedia-org.translate.goog/wiki/Tanjore_District_(Madras_Presidency)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Tanjore District (Madras Presidency)">Tanjore District</a>, <a href="https://en-m-wikipedia-org.translate.goog/wiki/Madras_Presidency?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Madras Presidency">Madras Presidency</a>, <a href="https://en-m-wikipedia-org.translate.goog/wiki/British_Raj?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="British Raj">British India</a> (now <a href="https://en-m-wikipedia-org.translate.goog/wiki/Thanjavur_district?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Thanjavur district">Thanjavur district</a>, <a href="https://en-m-wikipedia-org.translate.goog/wiki/Tamil_Nadu?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Tamil Nadu">Tamil Nadu</a>, <a href="https://en-m-wikipedia-org.translate.goog/wiki/India?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="India">India</a>) </div></td> </tr> <tr> <th scope="row" class="infobox-label">Citizenship</th> <td class="infobox-data category"><a href="https://en-m-wikipedia-org.translate.goog/wiki/British_Raj?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="British Raj">British Indian</a></td> </tr> <tr> <th scope="row" class="infobox-label">Education</th> <td class="infobox-data"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="plainlist"> <ul> <li><a href="https://en-m-wikipedia-org.translate.goog/wiki/Government_Arts_College,_Kumbakonam?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Government Arts College, Kumbakonam">Government Arts College</a></li> <li><a href="https://en-m-wikipedia-org.translate.goog/wiki/Pachaiyappa%27s_College?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Pachaiyappa's College">Pachaiyappa's College</a></li> <li><a href="https://en-m-wikipedia-org.translate.goog/wiki/Trinity_College,_Cambridge?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Trinity College, Cambridge">Trinity College, Cambridge</a> (<a href="https://en-m-wikipedia-org.translate.goog/wiki/Bachelor_of_Arts?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Bachelor of Arts">BA</a>)</li> </ul> </div></td> </tr> <tr> <th scope="row" class="infobox-label">Known for</th> <td class="infobox-data"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"> <div class="plainlist"> <ul> <li><a href="https://en-m-wikipedia-org.translate.goog/wiki/Ramanujan%27s_sum?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Ramanujan's sum">Ramanujan's sum</a></li> <li><a href="https://en-m-wikipedia-org.translate.goog/wiki/Landau%E2%80%93Ramanujan_constant?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Landau–Ramanujan constant">Landau–Ramanujan constant</a></li> <li><a href="https://en-m-wikipedia-org.translate.goog/wiki/Mock_theta_function?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Mock theta function">Mock theta functions</a></li> <li><a href="https://en-m-wikipedia-org.translate.goog/wiki/Ramanujan_conjecture?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Ramanujan conjecture">Ramanujan conjecture</a></li> <li><a href="https://en-m-wikipedia-org.translate.goog/wiki/Ramanujan_prime?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Ramanujan prime">Ramanujan prime</a></li> <li><a href="https://en-m-wikipedia-org.translate.goog/wiki/Ramanujan%E2%80%93Soldner_constant?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Ramanujan–Soldner constant">Ramanujan–Soldner constant</a></li> <li><a href="https://en-m-wikipedia-org.translate.goog/wiki/Ramanujan_theta_function?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Ramanujan theta function">Ramanujan theta function</a></li> <li><a href="https://en-m-wikipedia-org.translate.goog/wiki/Rogers%E2%80%93Ramanujan_identities?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Rogers–Ramanujan identities">Rogers–Ramanujan identities</a></li> <li><a href="https://en-m-wikipedia-org.translate.goog/wiki/Ramanujan%27s_master_theorem?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Ramanujan's master theorem">Ramanujan's master theorem</a></li> <li><a href="https://en-m-wikipedia-org.translate.goog/wiki/Partition_function_(number_theory)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Approximation_formulas" title="Partition function (number theory)">Hardy–Ramanujan asymptotic formula</a></li> <li><a href="https://en-m-wikipedia-org.translate.goog/wiki/Ramanujan%E2%80%93Sato_series?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Ramanujan–Sato series">Ramanujan–Sato series</a></li> </ul> </div></td> </tr> <tr> <th scope="row" class="infobox-label">Awards</th> <td class="infobox-data"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Fellow_of_the_Royal_Society?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Fellow of the Royal Society">Fellow of the Royal Society</a> (1918)</td> </tr> <tr> <td colspan="2" class="infobox-full-data"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1257001546">Scientific career</td> </tr> <tr> <th scope="row" class="infobox-label">Fields</th> <td class="infobox-data category"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Mathematics?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Mathematics">Mathematics</a></td> </tr> <tr> <th scope="row" class="infobox-label">Institutions</th> <td class="infobox-data"><a href="https://en-m-wikipedia-org.translate.goog/wiki/University_of_Cambridge?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="University of Cambridge">University of Cambridge</a></td> </tr> <tr> <th scope="row" class="infobox-label"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Thesis?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Thesis">Thesis</a></th> <td class="infobox-data"><a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://web.archive.org/web/20180809110104/http://math.univ-lyon1.fr/~nicolas/ramanujanNR.pdf">Highly Composite Numbers</a> (1916)</td> </tr> <tr> <th scope="row" class="infobox-label">Academic advisors</th> <td class="infobox-data"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"> <div class="plainlist"> <ul> <li><a href="https://en-m-wikipedia-org.translate.goog/wiki/G._H._Hardy?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="G. H. Hardy">G. H. Hardy</a></li> <li><a href="https://en-m-wikipedia-org.translate.goog/wiki/John_Edensor_Littlewood?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="John Edensor Littlewood">J. E. Littlewood</a></li> </ul> </div></td> </tr> <tr style="display:none"> <td colspan="2"></td> </tr> <tr> <th colspan="2" class="infobox-header">Signature</th> </tr> <tr> <td colspan="2" class="infobox-full-data"><a href="https://en-m-wikipedia-org.translate.goog/wiki/File:Signature_of_Srinivasa_Ramanujan.svg?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-file-description"><img alt="Srinivasa Ramanujan signature" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/10/Signature_of_Srinivasa_Ramanujan.svg/150px-Signature_of_Srinivasa_Ramanujan.svg.png" decoding="async" width="150" height="36" class="mw-file-element" srcset="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://upload.wikimedia.org/wikipedia/commons/thumb/1/10/Signature_of_Srinivasa_Ramanujan.svg/225px-Signature_of_Srinivasa_Ramanujan.svg.png 1.5x,https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://upload.wikimedia.org/wikipedia/commons/thumb/1/10/Signature_of_Srinivasa_Ramanujan.svg/300px-Signature_of_Srinivasa_Ramanujan.svg.png 2x" data-file-width="2500" data-file-height="593"></a></td> </tr> </tbody> </table> Ramanujan initially developed his own mathematical research in isolation. According to <a href="https://en-m-wikipedia-org.translate.goog/wiki/Hans_Eysenck?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Hans Eysenck">Hans Eysenck</a>, "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".<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-5">[4]</a> Seeking mathematicians who could better understand his work, in 1913 he began a <a href="https://en-m-wikipedia-org.translate.goog/wiki/Mail?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Mail">mail</a> correspondence with the English mathematician <a href="https://en-m-wikipedia-org.translate.goog/wiki/G._H._Hardy?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="G. H. Hardy">G. H. Hardy</a> at the <a href="https://en-m-wikipedia-org.translate.goog/wiki/University_of_Cambridge?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="University of Cambridge">University of Cambridge</a>, 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 <a href="https://en-m-wikipedia-org.translate.goog/wiki/Theorem?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Theorem">theorems</a>, including some that "defeated me completely; I had never seen anything in the least like them before",<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-6">[5]</a> and some recently proven but highly advanced results. During his short life, Ramanujan independently compiled nearly 3,900 results (mostly <a href="https://en-m-wikipedia-org.translate.goog/wiki/Identity_(mathematics)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Identity (mathematics)">identities</a> and <a href="https://en-m-wikipedia-org.translate.goog/wiki/Equation?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Equation">equations</a>).<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-7">[6]</a> Many were completely novel; his original and highly unconventional results, such as the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Ramanujan_prime?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Ramanujan prime">Ramanujan prime</a>, the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Ramanujan_theta_function?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Ramanujan theta function">Ramanujan theta function</a>, <a href="https://en-m-wikipedia-org.translate.goog/wiki/Partition_(number_theory)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Partition (number theory)">partition</a> formulae and <a href="https://en-m-wikipedia-org.translate.goog/wiki/Mock_theta_function?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Mock theta function">mock theta functions</a>, have opened entire new areas of work and inspired further research.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-ono-8">[7]</a> Of his thousands of results, most have been proven correct.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-frontline-9">[8]</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/The_Ramanujan_Journal?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="The Ramanujan Journal">The Ramanujan Journal</a>, a <a href="https://en-m-wikipedia-org.translate.goog/wiki/Scientific_journal?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Scientific journal">scientific journal</a>, was established to publish work in all areas of mathematics influenced by Ramanujan,<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-10">[9]</a> 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.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-ono_simpleproperties-11">[10]</a><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-ono_mock-12">[11]</a> He became one of the youngest <a href="https://en-m-wikipedia-org.translate.goog/wiki/Fellow_of_the_Royal_Society?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Fellow of the Royal Society">Fellows of the Royal Society</a> and only the second Indian member, and the first Indian to be elected a <a href="https://en-m-wikipedia-org.translate.goog/wiki/Trinity_College,_Cambridge?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Notable_fellows_and_alumni" title="Trinity College, Cambridge">Fellow of Trinity College, Cambridge</a>. In 1919, ill health—now believed to have been hepatic <a href="https://en-m-wikipedia-org.translate.goog/wiki/Amoebiasis?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Amoebiasis">amoebiasis</a> (a complication from episodes of <a href="https://en-m-wikipedia-org.translate.goog/wiki/Dysentery?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Dysentery">dysentery</a> 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 "<a href="https://en-m-wikipedia-org.translate.goog/wiki/Ramanujan%27s_lost_notebook?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Ramanujan's lost notebook">lost notebook</a>", containing discoveries from the last year of his life, caused great excitement among mathematicians when it was rediscovered in 1976. <div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"> <input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none"> <div class="toctitle" lang="en" dir="ltr"> <h2 id="mw-toc-heading">Contents</h2><label class="toctogglelabel" for="toctogglecheckbox"></label> </div> <ul> <li class="toclevel-1 tocsection-1"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Early_life">1 Early life</a></li> <li class="toclevel-1 tocsection-2"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Adulthood_in_India">2 Adulthood in India</a> <ul> <li class="toclevel-2 tocsection-3"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Pursuit_of_career_in_mathematics">2.1 Pursuit of career in mathematics</a></li> <li class="toclevel-2 tocsection-4"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Contacting_British_mathematicians">2.2 Contacting British mathematicians</a></li> </ul></li> <li class="toclevel-1 tocsection-5"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Life_in_England">3 Life in England</a></li> <li class="toclevel-1 tocsection-6"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Illness_and_death">4 Illness and death</a></li> <li class="toclevel-1 tocsection-7"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Personality_and_spiritual_life">5 Personality and spiritual life</a></li> <li class="toclevel-1 tocsection-8"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Mathematical_achievements">6 Mathematical achievements</a> <ul> <li class="toclevel-2 tocsection-9"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#The_Ramanujan_conjecture">6.1 The Ramanujan conjecture</a></li> <li class="toclevel-2 tocsection-10"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Ramanujan's_notebooks">6.2 Ramanujan's notebooks</a></li> </ul></li> <li class="toclevel-1 tocsection-11"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Hardy%E2%80%93Ramanujan_number_1729">7 Hardy–Ramanujan number 1729</a></li> <li class="toclevel-1 tocsection-12"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Mathematicians'_views_of_Ramanujan">8 Mathematicians' views of Ramanujan</a></li> <li class="toclevel-1 tocsection-13"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Posthumous_recognition">9 Posthumous recognition</a></li> <li class="toclevel-1 tocsection-14"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Commemorative_postal_stamps">10 Commemorative postal stamps</a></li> <li class="toclevel-1 tocsection-15"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#In_popular_culture">11 In popular culture</a></li> <li class="toclevel-1 tocsection-16"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Selected_papers">12 Selected papers</a></li> <li class="toclevel-1 tocsection-17"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Further_works_of_Ramanujan's_mathematics">13 Further works of Ramanujan's mathematics</a></li> <li class="toclevel-1 tocsection-18"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Selected_publications_on_Ramanujan_and_his_work">14 Selected publications on Ramanujan and his work</a></li> <li class="toclevel-1 tocsection-19"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Selected_publications_on_works_of_Ramanujan">15 Selected publications on works of Ramanujan</a></li> <li class="toclevel-1 tocsection-20"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#See_also">16 See also</a></li> <li class="toclevel-1 tocsection-21"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Footnotes">17 Footnotes</a></li> <li class="toclevel-1 tocsection-22"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#References">18 References</a></li> <li class="toclevel-1 tocsection-23"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#External_links">19 External links</a> <ul> <li class="toclevel-2 tocsection-24"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Media_links">19.1 Media links</a></li> <li class="toclevel-2 tocsection-25"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Biographical_links">19.2 Biographical links</a></li> <li class="toclevel-2 tocsection-26"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Other_links">19.3 Other links</a></li> </ul></li> </ul> </div> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(1)"> <h2 id="Early_life">Early life</h2> </div> <section class="mf-section-1 collapsible-block" id="mf-section-1"> <figure class="mw-default-size" typeof="mw:File/Thumb"> <a href="https://en-m-wikipedia-org.translate.goog/wiki/File:Erode,_18_Alahiri_Str_-_Ramanujan_birth_place.jpg?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-file-description"> <noscript> <img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fd/Erode%2C_18_Alahiri_Str_-_Ramanujan_birth_place.jpg/170px-Erode%2C_18_Alahiri_Str_-_Ramanujan_birth_place.jpg" decoding="async" width="170" height="227" class="mw-file-element" data-file-width="1920" data-file-height="2560"> </noscript><span class="lazy-image-placeholder" style="width: 170px;height: 227px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fd/Erode%2C_18_Alahiri_Str_-_Ramanujan_birth_place.jpg/170px-Erode%2C_18_Alahiri_Str_-_Ramanujan_birth_place.jpg" data-width="170" data-height="227" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fd/Erode%2C_18_Alahiri_Str_-_Ramanujan_birth_place.jpg/255px-Erode%2C_18_Alahiri_Str_-_Ramanujan_birth_place.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fd/Erode%2C_18_Alahiri_Str_-_Ramanujan_birth_place.jpg/340px-Erode%2C_18_Alahiri_Str_-_Ramanujan_birth_place.jpg 2x" data-class="mw-file-element"> </a> <figcaption> Ramanujan's birthplace on 18 Alahiri Street, <a href="https://en-m-wikipedia-org.translate.goog/wiki/Erode?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Erode">Erode</a>, now in <a href="https://en-m-wikipedia-org.translate.goog/wiki/Tamil_Nadu?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Tamil Nadu">Tamil Nadu</a> </figcaption> </figure> <figure class="mw-default-size" typeof="mw:File/Thumb"> <a href="https://en-m-wikipedia-org.translate.goog/wiki/File:Ramanujanhome.jpg?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-file-description"> <noscript> <img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/31/Ramanujanhome.jpg/170px-Ramanujanhome.jpg" decoding="async" width="170" height="227" class="mw-file-element" data-file-width="2304" data-file-height="3072"> </noscript><span class="lazy-image-placeholder" style="width: 170px;height: 227px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/3/31/Ramanujanhome.jpg/170px-Ramanujanhome.jpg" data-width="170" data-height="227" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/31/Ramanujanhome.jpg/255px-Ramanujanhome.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/31/Ramanujanhome.jpg/340px-Ramanujanhome.jpg 2x" data-class="mw-file-element"> </a> <figcaption> Ramanujan's home on Sarangapani Sannidhi Street, <a href="https://en-m-wikipedia-org.translate.goog/wiki/Kumbakonam?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Kumbakonam">Kumbakonam</a> </figcaption> </figure> Ramanujan (literally, "younger brother of <a href="https://en-m-wikipedia-org.translate.goog/wiki/Rama?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Rama">Rama</a>", a Hindu deity)<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Kanigel-1991-12-13">[12]</a> was born on 22 December 1887 into a <a href="https://en-m-wikipedia-org.translate.goog/wiki/Tamil_Brahmin?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Tamil Brahmin">Tamil Brahmin</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Iyengar?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Iyengar">Iyengar</a> family in <a href="https://en-m-wikipedia-org.translate.goog/wiki/Erode?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Erode">Erode</a>, in present-day <a href="https://en-m-wikipedia-org.translate.goog/wiki/Tamil_Nadu?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Tamil Nadu">Tamil Nadu</a>.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-14">[13]</a> His father, Kuppuswamy Srinivasa Iyengar, originally from <a href="https://en-m-wikipedia-org.translate.goog/wiki/Thanjavur_district?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Thanjavur district">Thanjavur district</a>, worked as a clerk in a <a href="https://en-m-wikipedia-org.translate.goog/wiki/Sari?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Sari">sari</a> shop.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-15">[14]</a><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-ODNB-Ramanujan-2">[2]</a> His mother, Komalatammal, was a <a href="https://en-m-wikipedia-org.translate.goog/wiki/Housewife?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Housewife">housewife</a> and sang at a local temple.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-16">[15]</a> They lived in a small traditional home on Sarangapani Sannidhi Street in the town of <a href="https://en-m-wikipedia-org.translate.goog/wiki/Kumbakonam?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Kumbakonam">Kumbakonam</a>.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-hindu-17">[16]</a> 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 <a href="https://en-m-wikipedia-org.translate.goog/wiki/Smallpox?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Smallpox">smallpox</a>, 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 <a href="https://en-m-wikipedia-org.translate.goog/wiki/Kanchipuram?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Kanchipuram">Kanchipuram</a>, near Madras (now <a href="https://en-m-wikipedia-org.translate.goog/wiki/Chennai?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Chennai">Chennai</a>). His mother gave birth to two more children, in 1891 and 1894, both of whom died before their first birthdays.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Kanigel-1991-12-13">[12]</a> On 1 October 1892, Ramanujan was enrolled at the local school.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-18">[17]</a> After his maternal grandfather lost his job as a court official in Kanchipuram,<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-19">[18]</a> Ramanujan and his mother moved back to <a href="https://en-m-wikipedia-org.translate.goog/wiki/Kumbakonam?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Kumbakonam">Kumbakonam</a>, and he was enrolled in Kangayan Primary School.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Kanigel-1991-14-20">[19]</a> 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.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Kanigel-1991-14-20">[19]</a> 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 <a href="https://en-m-wikipedia-org.translate.goog/wiki/Puranas?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Puranas">puranas</a>, to sing religious songs, to attend <a href="https://en-m-wikipedia-org.translate.goog/wiki/Puja_(Hinduism)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Puja (Hinduism)">pujas</a> at the temple, and to maintain particular eating habits—all part of <a href="https://en-m-wikipedia-org.translate.goog/wiki/Brahmin?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Brahmin">Brahmin</a> culture.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-21">[20]</a> At Kangayan Primary School, Ramanujan performed well. Just before turning 10, in November 1897, he passed his primary examinations in English, <a href="https://en-m-wikipedia-org.translate.goog/wiki/Tamil_language?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Tamil language">Tamil</a>, geography, and arithmetic with the best scores in the district.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Kanigel-1991-25-22">[21]</a> That year, Ramanujan entered <a href="https://en-m-wikipedia-org.translate.goog/wiki/Town_Higher_Secondary_School?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Town Higher Secondary School">Town Higher Secondary School</a>, where he encountered formal mathematics for the first time.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Kanigel-1991-25-22">[21]</a> A <a href="https://en-m-wikipedia-org.translate.goog/wiki/Child_prodigy?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Child prodigy">child prodigy</a> 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 <a href="https://en-m-wikipedia-org.translate.goog/wiki/S._L._Loney?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="S. L. Loney">S. L. Loney</a> on advanced trigonometry.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-berndt9-23">[22]</a><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-24">[23]</a> 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.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Kanigel-1991-27-25">[24]</a> He completed mathematical exams in half the allotted time, and showed a familiarity with <a href="https://en-m-wikipedia-org.translate.goog/wiki/Geometry?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Geometry">geometry</a> and <a href="https://en-m-wikipedia-org.translate.goog/wiki/Infinite_series?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Infinite series">infinite series</a>. Ramanujan was shown how to solve cubic equations in 1902. He would later develop his own method to solve the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Quartic_function?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Quartic function">quartic</a>. In 1903, he tried to solve the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Quintic_function?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Quintic function">quintic</a>, not knowing that it was <a href="https://en-m-wikipedia-org.translate.goog/wiki/Abel%E2%80%93Ruffini_theorem?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Abel–Ruffini theorem">impossible to solve with radicals.</a><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-26">[25]</a> In 1903, when he was 16, Ramanujan obtained from a friend a library copy of <a href="https://en-m-wikipedia-org.translate.goog/wiki/Synopsis_of_Pure_Mathematics?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Synopsis of Pure Mathematics">A Synopsis of Elementary Results in Pure and Applied Mathematics</a>, <a href="https://en-m-wikipedia-org.translate.goog/wiki/G._S._Carr?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="G. S. Carr">G. S. Carr</a>'s collection of 5,000 theorems.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-27">[26]</a><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-28">[27]</a> Ramanujan reportedly studied the contents of the book in detail.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-papers-29">[28]</a> The next year, Ramanujan independently developed and investigated the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Bernoulli_number?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Bernoulli number">Bernoulli numbers</a> and calculated the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Euler%E2%80%93Mascheroni_constant?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Euler–Mascheroni constant">Euler–Mascheroni constant</a> up to 15 decimal places.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-30">[29]</a> His peers at the time said they "rarely understood him" and "stood in respectful awe" of him.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Kanigel-1991-27-25">[24]</a> 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.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Kanigel-1991-unknown-page-31">[30]</a> He received a scholarship to study at <a href="https://en-m-wikipedia-org.translate.goog/wiki/Government_Arts_College,_Kumbakonam?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Government Arts College, Kumbakonam">Government Arts College, Kumbakonam</a>,<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Kanigel-1991-28-32">[31]</a><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Kanigel-1991-45-33">[32]</a> 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.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Kanigel-1991-47-48-34">[33]</a> In August 1905, Ramanujan ran away from home, heading towards <a href="https://en-m-wikipedia-org.translate.goog/wiki/Visakhapatnam?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Visakhapatnam">Visakhapatnam</a>, and stayed in <a href="https://en-m-wikipedia-org.translate.goog/wiki/Rajahmundry?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Rajahmundry">Rajahmundry</a><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-35">[34]</a> for about a month.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Kanigel-1991-47-48-34">[33]</a> He later enrolled at <a href="https://en-m-wikipedia-org.translate.goog/wiki/Pachaiyappa%27s_College?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Pachaiyappa's College">Pachaiyappa's College</a> 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.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-hindu2-36">[35]</a> Ramanujan failed his <a href="https://en-m-wikipedia-org.translate.goog/wiki/Royal_Society_of_Arts?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Royal Society of Arts">Fellow of Arts</a> 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.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Kanigel-1991-55-56-37">[36]</a> In 1910, after a meeting between the 23-year-old Ramanujan and the founder of the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Indian_Mathematical_Society?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Indian Mathematical Society">Indian Mathematical Society</a>, <a href="https://en-m-wikipedia-org.translate.goog/wiki/V._Ramaswamy_Aiyer?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="V. Ramaswamy Aiyer">V. Ramaswamy Aiyer</a>, Ramanujan began to get recognition in Madras's mathematical circles, leading to his inclusion as a researcher at the <a href="https://en-m-wikipedia-org.translate.goog/wiki/University_of_Madras?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="University of Madras">University of Madras</a>.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-krish-38">[37]</a> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(2)"> <h2 id="Adulthood_in_India">Adulthood in India</h2> </div> <section class="mf-section-2 collapsible-block" id="mf-section-2"> On 14 July 1909, Ramanujan married Janaki (Janakiammal; 21 March 1899 – 13 April 1994),<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-39">[38]</a> a girl his mother had selected for him a year earlier and who was ten years old when they married.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Kanigel-1991-71-40">[39]</a><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-History_in_adult_human_sexual_behavior_with_children_and_adolescents_in_Western_societies-41">[40]</a><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-42">[41]</a> 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 (<a href="https://en-m-wikipedia-org.translate.goog/wiki/Karur_district?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Karur district">Karur district</a>) Railway Station. Ramanujan's father did not participate in the marriage ceremony.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Janaki-43">[42]</a> 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.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-express-44">[43]</a> <figure class="mw-default-size" typeof="mw:File/Thumb"> <a href="https://en-m-wikipedia-org.translate.goog/wiki/File:Ramanujan_seated_alone.png?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-file-description"> <noscript> <img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c4/Ramanujan_seated_alone.png/220px-Ramanujan_seated_alone.png" decoding="async" width="220" height="415" class="mw-file-element" data-file-width="280" data-file-height="528"> </noscript><span class="lazy-image-placeholder" style="width: 220px;height: 415px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c4/Ramanujan_seated_alone.png/220px-Ramanujan_seated_alone.png" data-width="220" data-height="415" data-srcset="//upload.wikimedia.org/wikipedia/commons/c/c4/Ramanujan_seated_alone.png 1.5x" data-class="mw-file-element"> </a> <figcaption> Ramanujan seated alone </figcaption> </figure> After the marriage, Ramanujan developed a <a href="https://en-m-wikipedia-org.translate.goog/wiki/Hydrocele_testis?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Hydrocele testis">hydrocele testis</a>.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Kanigel-1991-72-45">[44]</a> 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.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-46">[45]</a> 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.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-47">[46]</a> 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 <a href="https://en-m-wikipedia-org.translate.goog/wiki/Madras_Christian_College?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Madras Christian College">Madras Christian College</a>."<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-48">[47]</a> After Ramanujan recovered and retrieved his notebooks from Iyer, he took a train from Kumbakonam to <a href="https://en-m-wikipedia-org.translate.goog/wiki/Viluppuram_(city)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Viluppuram (city)">Villupuram</a>, a city under French control.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-49">[48]</a><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-50">[49]</a> In 1912, Ramanujan moved with his wife and mother to a house in Saiva Muthaiah Mudali street, <a href="https://en-m-wikipedia-org.translate.goog/wiki/George_Town,_Chennai?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="George Town, Chennai">George Town</a>, <a href="https://en-m-wikipedia-org.translate.goog/wiki/Madras?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Madras">Madras</a>, where they lived for a few months.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-rao-51">[50]</a> In May 1913, upon securing a research position at Madras University, Ramanujan moved with his family to <a href="https://en-m-wikipedia-org.translate.goog/wiki/Triplicane?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Triplicane">Triplicane</a>.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-52">[51]</a> <div class="mw-heading mw-heading3"> <h3 id="Pursuit_of_career_in_mathematics">Pursuit of career in mathematics</h3> </div> In 1910, Ramanujan met deputy collector <a href="https://en-m-wikipedia-org.translate.goog/wiki/V._Ramaswamy_Aiyer?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="V. Ramaswamy Aiyer">V. Ramaswamy Aiyer</a>, who founded the Indian Mathematical Society.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Kanigel-1991-77-53">[52]</a> 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.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-54">[53]</a> </blockquote> Aiyer sent Ramanujan, with letters of introduction, to his mathematician friends in Madras.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Kanigel-1991-77-53">[52]</a> Some of them looked at his work and gave him letters of introduction to <a href="https://en-m-wikipedia-org.translate.goog/wiki/R._Ramachandra_Rao?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="R. Ramachandra Rao">R. Ramachandra Rao</a>, the district collector for <a href="https://en-m-wikipedia-org.translate.goog/wiki/Nellore?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Nellore">Nellore</a> and the secretary of the Indian Mathematical Society.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-55">[54]</a><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-56">[55]</a><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-57">[56]</a> 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 <a href="https://en-m-wikipedia-org.translate.goog/wiki/Mumbai?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Mumbai">Bombay</a> mathematician, in which Saldhana expressed a lack of understanding of his work but concluded that he was not a fraud.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Kanigel-1991-80-58">[57]</a> 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 <a href="https://en-m-wikipedia-org.translate.goog/wiki/Elliptic_integral?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Elliptic integral">elliptic integrals</a>, <a href="https://en-m-wikipedia-org.translate.goog/wiki/Gaussian_hypergeometric_series?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Gaussian hypergeometric series">hypergeometric series</a>, and his theory of <a href="https://en-m-wikipedia-org.translate.goog/wiki/Divergent_series?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Divergent series">divergent series</a>, which Rao said ultimately convinced him of Ramanujan's brilliance.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Kanigel-1991-80-58">[57]</a> 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.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-59">[58]</a> <figure class="mw-default-size" typeof="mw:File/Thumb"> <a href="https://en-m-wikipedia-org.translate.goog/wiki/File:K_Ananda_Rau_seated_with_Ramanujan.jpg?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-file-description"> <noscript> <img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/75/K_Ananda_Rau_seated_with_Ramanujan.jpg/220px-K_Ananda_Rau_seated_with_Ramanujan.jpg" decoding="async" width="220" height="242" class="mw-file-element" data-file-width="443" data-file-height="488"> </noscript><span class="lazy-image-placeholder" style="width: 220px;height: 242px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/7/75/K_Ananda_Rau_seated_with_Ramanujan.jpg/220px-K_Ananda_Rau_seated_with_Ramanujan.jpg" data-width="220" data-height="242" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/75/K_Ananda_Rau_seated_with_Ramanujan.jpg/330px-K_Ananda_Rau_seated_with_Ramanujan.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/75/K_Ananda_Rau_seated_with_Ramanujan.jpg/440px-K_Ananda_Rau_seated_with_Ramanujan.jpg 2x" data-class="mw-file-element"> </a> <figcaption> K Ananda Rau seated with Ramanujan </figcaption> </figure> One of the first problems he posed in the journal<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Kanigel-1991-unknown-page-31">[30]</a> was to find the value of: <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {1+2{\sqrt {1+3{\sqrt {1+\cdots }}}}}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn> 1 </mn> <mo> + </mo> <mn> 2 </mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn> 1 </mn> <mo> + </mo> <mn> 3 </mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn> 1 </mn> <mo> + </mo> <mo> ⋯ </mo> </msqrt> </mrow> </msqrt> </mrow> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\sqrt {1+2{\sqrt {1+3{\sqrt {1+\cdots }}}}}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c777a0f8cdb7faf0321155dacb2b1b073684e5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:23.64ex; height:6.176ex;" alt="{\displaystyle {\sqrt {1+2{\sqrt {1+3{\sqrt {1+\cdots }}}}}}}"> </noscript><span class="lazy-image-placeholder" style="width: 23.64ex;height: 6.176ex;vertical-align: -2.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c777a0f8cdb7faf0321155dacb2b1b073684e5c" data-alt="{\displaystyle {\sqrt {1+2{\sqrt {1+3{\sqrt {1+\cdots }}}}}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> 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<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-60">[59]</a> solution to the problem himself. On page 105 of his first notebook, he formulated an equation that could be used to solve the infinitely <a href="https://en-m-wikipedia-org.translate.goog/wiki/Nested_radical?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Nested radical">nested radicals</a> problem. <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x+n+a={\sqrt {ax+(n+a)^{2}+x{\sqrt {a(x+n)+(n+a)^{2}+(x+n){\sqrt {\cdots }}}}}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> x </mi> <mo> + </mo> <mi> n </mi> <mo> + </mo> <mi> a </mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi> a </mi> <mi> x </mi> <mo> + </mo> <mo stretchy="false"> ( </mo> <mi> n </mi> <mo> + </mo> <mi> a </mi> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi> a </mi> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo> + </mo> <mi> n </mi> <mo stretchy="false"> ) </mo> <mo> + </mo> <mo stretchy="false"> ( </mo> <mi> n </mi> <mo> + </mo> <mi> a </mi> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo> + </mo> <mi> n </mi> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo> ⋯ </mo> </msqrt> </mrow> </msqrt> </mrow> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle x+n+a={\sqrt {ax+(n+a)^{2}+x{\sqrt {a(x+n)+(n+a)^{2}+(x+n){\sqrt {\cdots }}}}}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83adced7a78dd85bae35d2ee34b2751989b9783e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:69.926ex; height:6.176ex;" alt="{\displaystyle x+n+a={\sqrt {ax+(n+a)^{2}+x{\sqrt {a(x+n)+(n+a)^{2}+(x+n){\sqrt {\cdots }}}}}}}"> </noscript><span class="lazy-image-placeholder" style="width: 69.926ex;height: 6.176ex;vertical-align: -2.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83adced7a78dd85bae35d2ee34b2751989b9783e" data-alt="{\displaystyle x+n+a={\sqrt {ax+(n+a)^{2}+x{\sqrt {a(x+n)+(n+a)^{2}+(x+n){\sqrt {\cdots }}}}}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> Using this equation, the answer to the question posed in the Journal was simply 3, obtained by setting x = 2, n = 1, and a = 0.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-61">[60]</a> Ramanujan wrote his first formal paper for the Journal on the properties of <a href="https://en-m-wikipedia-org.translate.goog/wiki/Bernoulli_number?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Bernoulli number">Bernoulli numbers</a>. One property he discovered was that the denominators of the fractions of Bernoulli numbers (sequence <a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://oeis.org/A027642" class="extiw" title="oeis:A027642">A027642</a> in the <a href="https://en-m-wikipedia-org.translate.goog/wiki/On-Line_Encyclopedia_of_Integer_Sequences?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>) are always divisible by six. He also devised a method of calculating Bn based on previous Bernoulli numbers. One of these methods follows: It will be observed that if n is even but not equal to zero, <ol> <li>Bn is a fraction and the numerator of <style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style>⁠Bn/n⁠ in its lowest terms is a prime number,</li> <li>the denominator of Bn contains each of the factors 2 and 3 once and only once,</li> <li>2n(2n − 1) <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠Bn/n⁠ is an integer and 2(2n − 1)Bn consequently is an odd integer.</li> </ol> In his 17-page paper "Some Properties of Bernoulli's Numbers" (1911), Ramanujan gave three proofs, two corollaries and three conjectures.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-62">[61]</a> 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.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-63">[62]</a> </blockquote> Ramanujan later wrote another paper and also continued to provide problems in the Journal.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Neville293-64">[63]</a> In early 1912, he got a temporary job in the Madras <a href="https://en-m-wikipedia-org.translate.goog/wiki/Accountant_General?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Accountant General">Accountant General</a>'s office, with a monthly salary of 20 rupees. He lasted only a few weeks.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-65">[64]</a> Toward the end of that assignment, he applied for a position under the Chief Accountant of the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Chennai_Port?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Chennai Port">Madras Port Trust</a>. In a letter dated 9 February 1912, Ramanujan wrote: <blockquote> Sir,  I understand there is a clerkship vacant in your office, and I beg to apply for the same. I have passed the Matriculation Examination and studied up to the F.A. but was prevented from pursuing my studies further owing to several untoward circumstances. I have, however, been devoting all my time to Mathematics and developing the subject. I can say I am quite confident I can do justice to my work if I am appointed to the post. I therefore beg to request that you will be good enough to confer the appointment on me.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-66">[65]</a> </blockquote> Attached to his application was a recommendation from <a href="https://en-m-wikipedia-org.translate.goog/wiki/E._W._Middlemast?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="E. W. Middlemast">E. W. Middlemast</a>, a mathematics professor at the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Presidency_College,_Chennai?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Presidency College, Chennai">Presidency College</a>, who wrote that Ramanujan was "a young man of quite exceptional capacity in Mathematics".<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-67">[66]</a> 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.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-68">[67]</a> At his office, Ramanujan easily and quickly completed the work he was given and spent his spare time doing mathematical research. Ramanujan's boss, <a href="https://en-m-wikipedia-org.translate.goog/wiki/Francis_Spring?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Francis Spring">Sir Francis Spring</a>, and S. Narayana Iyer, a colleague who was also treasurer of the Indian Mathematical Society, encouraged Ramanujan in his mathematical pursuits.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-FOOTNOTEBerndtRankin200197-69">[68]</a> <div class="mw-heading mw-heading3"> <h3 id="Contacting_British_mathematicians">Contacting British mathematicians</h3> </div> In the spring of 1913, Narayana Iyer, Ramachandra Rao and E. W. Middlemast tried to present Ramanujan's work to British mathematicians. <a href="https://en-m-wikipedia-org.translate.goog/wiki/Micaiah_John_Muller_Hill?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Micaiah John Muller Hill">M. J. M. Hill</a> of <a href="https://en-m-wikipedia-org.translate.goog/wiki/University_College_London?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="University College London">University College London</a> commented that Ramanujan's papers were riddled with holes.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-70">[69]</a> 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.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-71">[70]</a> 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.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-72">[71]</a> The first two professors, <a href="https://en-m-wikipedia-org.translate.goog/wiki/H._F._Baker?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="H. F. Baker">H. F. Baker</a> and <a href="https://en-m-wikipedia-org.translate.goog/wiki/E._W._Hobson?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="E. W. Hobson">E. W. Hobson</a>, returned Ramanujan's papers without comment.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-73">[72]</a> On 16 January 1913, Ramanujan wrote to <a href="https://en-m-wikipedia-org.translate.goog/wiki/G._H._Hardy?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="G. H. Hardy">G. H. Hardy</a>, whom he knew from studying Orders of Infinity (1910).<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-74">[73]</a><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-75">[74]</a> Coming from an unknown mathematician, the nine pages of mathematics made Hardy initially view Ramanujan's manuscripts as a possible fraud.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-76">[75]</a> Hardy recognised some of Ramanujan's formulae but others "seemed scarcely possible to believe".<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Hardy1920-77">[76]</a>: 494 One of the theorems Hardy found amazing was on the bottom of page three (valid for 0 < a < b + <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠): <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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+{\frac {1}{2}}\right)\Gamma (b+1)\Gamma (b-a+1)}{\Gamma (a)\Gamma \left(b+{\frac {1}{2}}\right)\Gamma \left(b-a+{\frac {1}{2}}\right)}}.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ∞ </mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <msup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mrow> <mo stretchy="false"> ( </mo> <mi> b </mi> <mo> + </mo> <mn> 1 </mn> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mrow> </mfrac> </mstyle> </mrow> </mrow> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <msup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <msup> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mfrac> </mstyle> </mrow> </mrow> </mfrac> </mrow> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <msup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mrow> <mo stretchy="false"> ( </mo> <mi> b </mi> <mo> + </mo> <mn> 2 </mn> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mrow> </mfrac> </mstyle> </mrow> </mrow> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <msup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mrow> <mo stretchy="false"> ( </mo> <mi> a </mi> <mo> + </mo> <mn> 1 </mn> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mrow> </mfrac> </mstyle> </mrow> </mrow> </mfrac> </mrow> <mo> × </mo> <mo> ⋯ </mo> <mspace width="thinmathspace"></mspace> <mi> d </mi> <mi> x </mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mi> π </mi> </msqrt> <mn> 2 </mn> </mfrac> </mrow> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal"> Γ </mi> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> <mi mathvariant="normal"> Γ </mi> <mo stretchy="false"> ( </mo> <mi> b </mi> <mo> + </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> <mi mathvariant="normal"> Γ </mi> <mo stretchy="false"> ( </mo> <mi> b </mi> <mo> − </mo> <mi> a </mi> <mo> + </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> </mrow> <mrow> <mi mathvariant="normal"> Γ </mi> <mo stretchy="false"> ( </mo> <mi> a </mi> <mo stretchy="false"> ) </mo> <mi mathvariant="normal"> Γ </mi> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> <mi mathvariant="normal"> Γ </mi> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> − </mo> <mi> a </mi> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \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+{\frac {1}{2}}\right)\Gamma (b+1)\Gamma (b-a+1)}{\Gamma (a)\Gamma \left(b+{\frac {1}{2}}\right)\Gamma \left(b-a+{\frac {1}{2}}\right)}}.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8789e669917365f58b1d6aab5d50454e2b0eba35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.005ex; width:84.132ex; height:13.343ex;" alt="{\displaystyle \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+{\frac {1}{2}}\right)\Gamma (b+1)\Gamma (b-a+1)}{\Gamma (a)\Gamma \left(b+{\frac {1}{2}}\right)\Gamma \left(b-a+{\frac {1}{2}}\right)}}.}"> </noscript><span class="lazy-image-placeholder" style="width: 84.132ex;height: 13.343ex;vertical-align: -6.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8789e669917365f58b1d6aab5d50454e2b0eba35" data-alt="{\displaystyle \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+{\frac {1}{2}}\right)\Gamma (b+1)\Gamma (b-a+1)}{\Gamma (a)\Gamma \left(b+{\frac {1}{2}}\right)\Gamma \left(b-a+{\frac {1}{2}}\right)}}.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> Hardy was also impressed by some of Ramanujan's other work relating to infinite series: <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1-5\left({\frac {1}{2}}\right)^{3}+9\left({\frac {1\times 3}{2\times 4}}\right)^{3}-13\left({\frac {1\times 3\times 5}{2\times 4\times 6}}\right)^{3}+\cdots ={\frac {2}{\pi }}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn> 1 </mn> <mo> − </mo> <mn> 5 </mn> <msup> <mrow> <mo> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msup> <mo> + </mo> <mn> 9 </mn> <msup> <mrow> <mo> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn> 1 </mn> <mo> × </mo> <mn> 3 </mn> </mrow> <mrow> <mn> 2 </mn> <mo> × </mo> <mn> 4 </mn> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msup> <mo> − </mo> <mn> 13 </mn> <msup> <mrow> <mo> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn> 1 </mn> <mo> × </mo> <mn> 3 </mn> <mo> × </mo> <mn> 5 </mn> </mrow> <mrow> <mn> 2 </mn> <mo> × </mo> <mn> 4 </mn> <mo> × </mo> <mn> 6 </mn> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msup> <mo> + </mo> <mo> ⋯ </mo> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 2 </mn> <mi> π </mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle 1-5\left({\frac {1}{2}}\right)^{3}+9\left({\frac {1\times 3}{2\times 4}}\right)^{3}-13\left({\frac {1\times 3\times 5}{2\times 4\times 6}}\right)^{3}+\cdots ={\frac {2}{\pi }}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db6799aabf0a962cd83b722886623ed2ebc93cdb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:56.594ex; height:6.509ex;" alt="{\displaystyle 1-5\left({\frac {1}{2}}\right)^{3}+9\left({\frac {1\times 3}{2\times 4}}\right)^{3}-13\left({\frac {1\times 3\times 5}{2\times 4\times 6}}\right)^{3}+\cdots ={\frac {2}{\pi }}}"> </noscript><span class="lazy-image-placeholder" style="width: 56.594ex;height: 6.509ex;vertical-align: -2.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db6799aabf0a962cd83b722886623ed2ebc93cdb" data-alt="{\displaystyle 1-5\left({\frac {1}{2}}\right)^{3}+9\left({\frac {1\times 3}{2\times 4}}\right)^{3}-13\left({\frac {1\times 3\times 5}{2\times 4\times 6}}\right)^{3}+\cdots ={\frac {2}{\pi }}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1+9\left({\frac {1}{4}}\right)^{4}+17\left({\frac {1\times 5}{4\times 8}}\right)^{4}+25\left({\frac {1\times 5\times 9}{4\times 8\times 12}}\right)^{4}+\cdots ={\frac {2{\sqrt {2}}}{{\sqrt {\pi }}\,\Gamma ^{2}\left({\frac {3}{4}}\right)}}.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn> 1 </mn> <mo> + </mo> <mn> 9 </mn> <msup> <mrow> <mo> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 4 </mn> </mrow> </msup> <mo> + </mo> <mn> 17 </mn> <msup> <mrow> <mo> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn> 1 </mn> <mo> × </mo> <mn> 5 </mn> </mrow> <mrow> <mn> 4 </mn> <mo> × </mo> <mn> 8 </mn> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 4 </mn> </mrow> </msup> <mo> + </mo> <mn> 25 </mn> <msup> <mrow> <mo> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn> 1 </mn> <mo> × </mo> <mn> 5 </mn> <mo> × </mo> <mn> 9 </mn> </mrow> <mrow> <mn> 4 </mn> <mo> × </mo> <mn> 8 </mn> <mo> × </mo> <mn> 12 </mn> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 4 </mn> </mrow> </msup> <mo> + </mo> <mo> ⋯ </mo> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn> 2 </mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn> 2 </mn> </msqrt> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi> π </mi> </msqrt> </mrow> <mspace width="thinmathspace"></mspace> <msup> <mi mathvariant="normal"> Γ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mrow> <mo> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 3 </mn> <mn> 4 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle 1+9\left({\frac {1}{4}}\right)^{4}+17\left({\frac {1\times 5}{4\times 8}}\right)^{4}+25\left({\frac {1\times 5\times 9}{4\times 8\times 12}}\right)^{4}+\cdots ={\frac {2{\sqrt {2}}}{{\sqrt {\pi }}\,\Gamma ^{2}\left({\frac {3}{4}}\right)}}.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5dd99b27f6b92d8518688a2f5f87408aee029b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:69.217ex; height:8.676ex;" alt="{\displaystyle 1+9\left({\frac {1}{4}}\right)^{4}+17\left({\frac {1\times 5}{4\times 8}}\right)^{4}+25\left({\frac {1\times 5\times 9}{4\times 8\times 12}}\right)^{4}+\cdots ={\frac {2{\sqrt {2}}}{{\sqrt {\pi }}\,\Gamma ^{2}\left({\frac {3}{4}}\right)}}.}"> </noscript><span class="lazy-image-placeholder" style="width: 69.217ex;height: 8.676ex;vertical-align: -4.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5dd99b27f6b92d8518688a2f5f87408aee029b3" data-alt="{\displaystyle 1+9\left({\frac {1}{4}}\right)^{4}+17\left({\frac {1\times 5}{4\times 8}}\right)^{4}+25\left({\frac {1\times 5\times 9}{4\times 8\times 12}}\right)^{4}+\cdots ={\frac {2{\sqrt {2}}}{{\sqrt {\pi }}\,\Gamma ^{2}\left({\frac {3}{4}}\right)}}.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> The first result had already been determined by <a href="https://en-m-wikipedia-org.translate.goog/wiki/Gustav_Conrad_Bauer?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Gustav Conrad Bauer">G. Bauer</a> in 1859. The second was new to Hardy, and was derived from a class of functions called <a href="https://en-m-wikipedia-org.translate.goog/wiki/Hypergeometric_series?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Hypergeometric series">hypergeometric series</a>, which had first been researched by Euler and Gauss. Hardy found these results "much more intriguing" than Gauss's work on integrals.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Kanigel-1991-167-78">[77]</a> After seeing <a href="https://en-m-wikipedia-org.translate.goog/wiki/Rogers%E2%80%93Ramanujan_continued_fraction?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Rogers–Ramanujan continued fraction">Ramanujan's theorems on continued fractions</a> 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",<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Kanigel-1991-168-79">[78]</a> and that they "must be true, because, if they were not true, no one would have the imagination to invent them".<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Kanigel-1991-168-79">[78]</a> Hardy asked a colleague, <a href="https://en-m-wikipedia-org.translate.goog/wiki/John_Edensor_Littlewood?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="John Edensor Littlewood">J. E. Littlewood</a>, 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".<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Hardy1920-77">[76]</a>: 494–495 One colleague, <a href="https://en-m-wikipedia-org.translate.goog/wiki/Eric_Harold_Neville?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Eric Harold Neville">E. H. Neville</a>, 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".<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Neville293-64">[63]</a> 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".<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-80">[79]</a> 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.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-81">[80]</a> In accordance with his Brahmin upbringing, Ramanujan refused to leave his country to "<a href="https://en-m-wikipedia-org.translate.goog/wiki/Kala_pani_(taboo)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Kala pani (taboo)">go to a foreign land</a>", and his parents were also opposed for the same reason.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-82">[81]</a> Meanwhile, he sent Hardy a letter packed with theorems, writing, "I have found a friend in you who views my labour sympathetically."<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-83">[82]</a> To supplement Hardy's endorsement, <a href="https://en-m-wikipedia-org.translate.goog/wiki/Gilbert_Walker_(physicist)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Gilbert Walker (physicist)">Gilbert Walker</a>, a former mathematical lecturer at <a href="https://en-m-wikipedia-org.translate.goog/wiki/Trinity_College,_Cambridge?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Trinity College, Cambridge">Trinity College, Cambridge</a>, looked at Ramanujan's work and expressed amazement, urging the young man to spend time at Cambridge.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-84">[83]</a> 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".<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-85">[84]</a> The board agreed to grant Ramanujan a monthly research scholarship of 75 rupees for the next two years at the <a href="https://en-m-wikipedia-org.translate.goog/wiki/University_of_Madras?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="University of Madras">University of Madras</a>.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-86">[85]</a> 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 <a href="https://en-m-wikipedia-org.translate.goog/wiki/Edward_Burns_Ross?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edward Burns Ross">Edward B. Ross</a> of <a href="https://en-m-wikipedia-org.translate.goog/wiki/Madras_Christian_College?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Madras Christian College">Madras Christian College</a>, 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.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-87">[86]</a> 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.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-88">[87]</a> 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.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-89">[88]</a> 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".<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Neville293-64">[63]</a> Apparently, Ramanujan's mother had a vivid dream in which Ramanujan was surrounded by Europeans, and the family goddess, <a href="https://en-m-wikipedia-org.translate.goog/wiki/Namagiri_Thayar?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Namagiri Thayar">the deity of Namagiri</a>, commanded her "to stand no longer between her son and the fulfilment of his life's purpose".<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Neville293-64">[63]</a> On 17 March 1914, Ramanujan travelled to England by ship,<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-90">[89]</a> leaving his wife to stay with his parents in India.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-FOOTNOTEBerndtRankin200183%E2%80%9384-91">[90]</a> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(3)"> <h2 id="Life_in_England">Life in England</h2> </div> <section class="mf-section-3 collapsible-block" id="mf-section-3"> <figure class="mw-default-size" typeof="mw:File/Thumb"> <a href="https://en-m-wikipedia-org.translate.goog/wiki/File:RamanujanCambridge.jpg?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-file-description"> <noscript> <img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/57/RamanujanCambridge.jpg/290px-RamanujanCambridge.jpg" decoding="async" width="290" height="187" class="mw-file-element" data-file-width="1403" data-file-height="904"> </noscript><span class="lazy-image-placeholder" style="width: 290px;height: 187px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/5/57/RamanujanCambridge.jpg/290px-RamanujanCambridge.jpg" data-width="290" data-height="187" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/57/RamanujanCambridge.jpg/435px-RamanujanCambridge.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/57/RamanujanCambridge.jpg/580px-RamanujanCambridge.jpg 2x" data-class="mw-file-element"> </a> <figcaption> Ramanujan (centre) and his colleague <a href="https://en-m-wikipedia-org.translate.goog/wiki/G._H._Hardy?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="G. H. Hardy">G. H. Hardy</a> (rightmost), with other scientists, outside the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Senate_House,_Cambridge?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Senate House, Cambridge">Senate House, Cambridge</a>, c.1914–19 </figcaption> </figure> <figure class="mw-default-size" typeof="mw:File/Thumb"> <a href="https://en-m-wikipedia-org.translate.goog/wiki/File:Whewell%27s_Court,_Trinity_College,_Cambridge.jpg?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-file-description"> <noscript> <img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/77/Whewell%27s_Court%2C_Trinity_College%2C_Cambridge.jpg/170px-Whewell%27s_Court%2C_Trinity_College%2C_Cambridge.jpg" decoding="async" width="170" height="227" class="mw-file-element" data-file-width="480" data-file-height="640"> </noscript><span class="lazy-image-placeholder" style="width: 170px;height: 227px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/7/77/Whewell%27s_Court%2C_Trinity_College%2C_Cambridge.jpg/170px-Whewell%27s_Court%2C_Trinity_College%2C_Cambridge.jpg" data-width="170" data-height="227" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/77/Whewell%27s_Court%2C_Trinity_College%2C_Cambridge.jpg/255px-Whewell%27s_Court%2C_Trinity_College%2C_Cambridge.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/77/Whewell%27s_Court%2C_Trinity_College%2C_Cambridge.jpg/340px-Whewell%27s_Court%2C_Trinity_College%2C_Cambridge.jpg 2x" data-class="mw-file-element"> </a> <figcaption> Whewell's Court, <a href="https://en-m-wikipedia-org.translate.goog/wiki/Trinity_College,_Cambridge?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Trinity College, Cambridge">Trinity College, Cambridge</a> </figcaption> </figure> Ramanujan departed from Madras aboard the S.S. Nevasa on 17 March 1914.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-92">[91]</a><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-93">[92]</a> 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.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-94">[93]</a> <figure class="mw-default-size" typeof="mw:File/Thumb"> <a href="https://en-m-wikipedia-org.translate.goog/wiki/File:Ramanujan%27s_%22Master_Theorem%22_page.jpg?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-file-description"> <noscript> <img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Ramanujan%27s_%22Master_Theorem%22_page.jpg/220px-Ramanujan%27s_%22Master_Theorem%22_page.jpg" decoding="async" width="220" height="287" class="mw-file-element" data-file-width="450" data-file-height="588"> </noscript><span class="lazy-image-placeholder" style="width: 220px;height: 287px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Ramanujan%27s_%22Master_Theorem%22_page.jpg/220px-Ramanujan%27s_%22Master_Theorem%22_page.jpg" data-width="220" data-height="287" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Ramanujan%27s_%22Master_Theorem%22_page.jpg/330px-Ramanujan%27s_%22Master_Theorem%22_page.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Ramanujan%27s_%22Master_Theorem%22_page.jpg/440px-Ramanujan%27s_%22Master_Theorem%22_page.jpg 2x" data-class="mw-file-element"> </a> <figcaption> Ramanujan's "Master Theorem" page </figcaption> </figure> <a href="https://en-m-wikipedia-org.translate.goog/wiki/G._H._Hardy?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="G. H. Hardy">Hardy</a> and <a href="https://en-m-wikipedia-org.translate.goog/wiki/J._E._Littlewood?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="J. E. Littlewood">Littlewood</a> 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.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-95">[94]</a> Ramanujan left a deep impression on Hardy and Littlewood. Littlewood commented, "I can believe that he's at least a <a href="https://en-m-wikipedia-org.translate.goog/wiki/Carl_Gustav_Jacob_Jacobi?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Carl Gustav Jacob Jacobi">Jacobi</a>",<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-96">[95]</a> while Hardy said he "can compare him only with <a href="https://en-m-wikipedia-org.translate.goog/wiki/Leonhard_Euler?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Leonhard Euler">Euler</a> or Jacobi."<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-97">[96]</a> Ramanujan spent nearly five years in <a href="https://en-m-wikipedia-org.translate.goog/wiki/Cambridge?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Cambridge">Cambridge</a> 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 <a href="https://en-m-wikipedia-org.translate.goog/wiki/Foundations_of_mathematics?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Foundations of mathematics">foundations of mathematics</a> had come into question and the need for <a href="https://en-m-wikipedia-org.translate.goog/wiki/Mathematical_rigor?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Mathematical rigor">mathematically rigorous</a> 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<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-98">[97]</a><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-99">[98]</a> (the predecessor of the PhD degree) in March 1916 for his work on <a href="https://en-m-wikipedia-org.translate.goog/wiki/Highly_composite_number?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Highly composite number">highly composite numbers</a>, sections of the first part of which had been published the preceding year in the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Proceedings_of_the_London_Mathematical_Society?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Proceedings of the London Mathematical Society">Proceedings of the London Mathematical Society</a>. 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'.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-100">[99]</a> On 6 December 1917, Ramanujan was elected to the London Mathematical Society. On 2 May 1918, he was elected a <a href="https://en-m-wikipedia-org.translate.goog/wiki/Fellow_of_the_Royal_Society?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Fellow of the Royal Society">Fellow of the Royal Society</a>,<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-FRS-Repository-101">[100]</a> the second Indian admitted, after <a href="https://en-m-wikipedia-org.translate.goog/wiki/Ardaseer_Cursetjee?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Ardaseer Cursetjee">Ardaseer Cursetjee</a> 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 <a href="https://en-m-wikipedia-org.translate.goog/wiki/Elliptic_function?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Elliptic function">elliptic functions</a> and the Theory of Numbers." On 13 October 1918, he was the first Indian to be elected a <a href="https://en-m-wikipedia-org.translate.goog/wiki/Trinity_College,_Cambridge?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Notable_fellows_and_alumni" title="Trinity College, Cambridge">Fellow of Trinity College, Cambridge</a>.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-102">[101]</a> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(4)"> <h2 id="Illness_and_death">Illness and death</h2> </div> <section class="mf-section-4 collapsible-block" id="mf-section-4"> 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 <a href="https://en-m-wikipedia-org.translate.goog/wiki/Rationing_in_the_United_Kingdom?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#First_World_War_1914%E2%80%931918" title="Rationing in the United Kingdom">wartime rationing in 1914–18</a>. He was diagnosed with <a href="https://en-m-wikipedia-org.translate.goog/wiki/Tuberculosis?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Tuberculosis">tuberculosis</a> and a severe <a href="https://en-m-wikipedia-org.translate.goog/wiki/Vitamin?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Vitamin">vitamin</a> deficiency, and confined to a <a href="https://en-m-wikipedia-org.translate.goog/wiki/Sanatorium?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Sanatorium">sanatorium</a>. He attempted suicide in late 1917 or early 1918 by jumping on the tracks of a London underground station. <a href="https://en-m-wikipedia-org.translate.goog/wiki/Scotland_Yard?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Scotland Yard">Scotland Yard</a> arrested him for attempting suicide (which was a crime), but released him after Hardy intervened.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-103">[102]</a><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-:1-104">[103]</a> In 1919, Ramanujan returned to <a href="https://en-m-wikipedia-org.translate.goog/wiki/Kumbakonam?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Kumbakonam">Kumbakonam</a>, <a href="https://en-m-wikipedia-org.translate.goog/wiki/Madras_Presidency?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Madras Presidency">Madras Presidency</a>, 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.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-express-44">[43]</a> In his last days, though in severe pain, "he continued doing his mathematics filling sheet after sheet with numbers", Janaki Ammal recounts.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-105">[104]</a> Ramanujan's widow, <a href="https://en-m-wikipedia-org.translate.goog/wiki/Smt.?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Smt.">Smt.</a> Janaki Ammal, moved to <a href="https://en-m-wikipedia-org.translate.goog/wiki/Bombay?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Bombay">Bombay</a>. In 1931, she returned to Madras and settled in <a href="https://en-m-wikipedia-org.translate.goog/wiki/Triplicane?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Triplicane">Triplicane</a>, 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 <a href="https://en-m-wikipedia-org.translate.goog/wiki/State_Bank_of_India?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="State Bank of India">State Bank of India</a> 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 <a href="https://en-m-wikipedia-org.translate.goog/wiki/Indian_National_Science_Academy?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Indian National Science Academy">Indian National Science Academy</a> and the state governments of <a href="https://en-m-wikipedia-org.translate.goog/wiki/Tamil_Nadu?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Tamil Nadu">Tamil Nadu</a>, <a href="https://en-m-wikipedia-org.translate.goog/wiki/Andhra_Pradesh?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Andhra Pradesh">Andhra Pradesh</a> and <a href="https://en-m-wikipedia-org.translate.goog/wiki/West_Bengal?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="West Bengal">West Bengal</a>. She continued to cherish Ramanujan's memory, and was active in efforts to increase his public recognition; prominent mathematicians, including George Andrews, <a href="https://en-m-wikipedia-org.translate.goog/wiki/Bruce_C._Berndt?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Bruce C. Berndt">Bruce C. Berndt</a> and <a href="https://en-m-wikipedia-org.translate.goog/wiki/B%C3%A9la_Bollob%C3%A1s?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Béla Bollobás">Béla Bollobás</a> made it a point to visit her while in India. She died at her Triplicane residence in 1994.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Janaki-43">[42]</a><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-express-44">[43]</a> A 1994 analysis of Ramanujan's medical records and symptoms by D. A. B. Young<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-:1-104">[103]</a> concluded that his medical <a href="https://en-m-wikipedia-org.translate.goog/wiki/Symptom?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Symptom">symptoms</a>—including his past relapses, fevers, and hepatic conditions—were much closer to those resulting from hepatic <a href="https://en-m-wikipedia-org.translate.goog/wiki/Amoebiasis?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Amoebiasis">amoebiasis</a>, an illness then widespread in Madras, than tuberculosis. He had two episodes of <a href="https://en-m-wikipedia-org.translate.goog/wiki/Dysentery?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Dysentery">dysentery</a> 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.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-lostnotebook-106">[105]</a> At the time, if properly diagnosed, amoebiasis was a treatable and often curable disease;<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-lostnotebook-106">[105]</a><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-107">[106]</a> British soldiers who contracted it during the First World War were being successfully cured of amoebiasis around the time Ramanujan left England.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-108">[107]</a> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(5)"> <h2 id="Personality_and_spiritual_life">Personality and spiritual life</h2> </div> <section class="mf-section-5 collapsible-block" id="mf-section-5"> <style data-mw-deduplicate="TemplateStyles:r1224211176">.mw-parser-output .quotebox{background-color:#F9F9F9;border:1px solid #aaa;box-sizing:border-box;padding:10px;font-size:88%;max-width:100%}.mw-parser-output .quotebox.floatleft{margin:.5em 1.4em .8em 0}.mw-parser-output .quotebox.floatright{margin:.5em 0 .8em 1.4em}.mw-parser-output .quotebox.centered{overflow:hidden;position:relative;margin:.5em auto .8em auto}.mw-parser-output .quotebox.floatleft span,.mw-parser-output .quotebox.floatright span{font-style:inherit}.mw-parser-output .quotebox>blockquote{margin:0;padding:0;border-left:0;font-family:inherit;font-size:inherit}.mw-parser-output .quotebox-title{text-align:center;font-size:110%;font-weight:bold}.mw-parser-output .quotebox-quote>:first-child{margin-top:0}.mw-parser-output .quotebox-quote:last-child>:last-child{margin-bottom:0}.mw-parser-output .quotebox-quote.quoted:before{font-family:"Times New Roman",serif;font-weight:bold;font-size:large;color:gray;content:" “ ";vertical-align:-45%;line-height:0}.mw-parser-output .quotebox-quote.quoted:after{font-family:"Times New Roman",serif;font-weight:bold;font-size:large;color:gray;content:" ” ";line-height:0}.mw-parser-output .quotebox .left-aligned{text-align:left}.mw-parser-output .quotebox .right-aligned{text-align:right}.mw-parser-output .quotebox .center-aligned{text-align:center}.mw-parser-output .quotebox .quote-title,.mw-parser-output .quotebox .quotebox-quote{display:block}.mw-parser-output .quotebox cite{display:block;font-style:normal}@media screen and (max-width:640px){.mw-parser-output .quotebox{width:100%!important;margin:0 0 .8em!important;float:none!important}}</style> <div class="quotebox pullquote floatright" style="width:30%; ;"> <blockquote class="quotebox-quote left-aligned" style=""> 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. </blockquote> <cite class="left-aligned" style="">—Srinivasa Ramanujan<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-109">[108]</a></cite> </div> Ramanujan has been described as a person of a somewhat shy and quiet disposition, a dignified man with pleasant manners.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-110">[109]</a> He lived a simple life at Cambridge.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-111">[110]</a> Ramanujan's first Indian biographers describe him as a rigorously <a href="https://en-m-wikipedia-org.translate.goog/wiki/Orthodox_Hinduism?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Orthodox Hinduism">orthodox Hindu</a>. He credited his acumen to his <a href="https://en-m-wikipedia-org.translate.goog/wiki/Kuladevata?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Kuladevata">family goddess</a>, <a href="https://en-m-wikipedia-org.translate.goog/wiki/Namagiri_Thayar?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Namagiri Thayar">Namagiri Thayar</a> (Goddess <a href="https://en-m-wikipedia-org.translate.goog/wiki/Lakshmi?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Lakshmi">Mahalakshmi</a>) of <a href="https://en-m-wikipedia-org.translate.goog/wiki/Namakkal?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Namakkal">Namakkal</a>. He looked to her for inspiration in his work<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-112">[111]</a> and said he dreamed of blood drops that symbolised her consort, <a href="https://en-m-wikipedia-org.translate.goog/wiki/Narasimha?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Narasimha">Narasimha</a>. Later he had visions of scrolls of complex mathematical content unfolding before his eyes.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Kanigel-1991-281-113">[112]</a> He often said, "An equation for me has no meaning unless it expresses a thought of God."<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-114">[113]</a> Hardy cites Ramanujan as remarking that all religions seemed equally true to him.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Kanigel-1991-283-115">[114]</a> 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 <a href="https://en-m-wikipedia-org.translate.goog/wiki/Vegetarian?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Vegetarian">vegetarianism</a>.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-berndt-116">[115]</a> 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.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-frontline-9">[8]</a> 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 <a href="https://en-m-wikipedia-org.translate.goog/wiki/Orthopraxy?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Orthopraxy">orthopraxy</a> was not in doubt.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-117">[116]</a> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(6)"> <h2 id="Mathematical_achievements">Mathematical achievements</h2> </div> <section class="mf-section-6 collapsible-block" id="mf-section-6"> 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. <a href="https://en-m-wikipedia-org.translate.goog/wiki/G._H._Hardy?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="G. H. Hardy">G. H. Hardy</a> 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 <a href="https://en-m-wikipedia-org.translate.goog/wiki/Series_(mathematics)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Series (mathematics)">series</a> for <a href="https://en-m-wikipedia-org.translate.goog/wiki/Pi?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Pi">π</a>, one of which is given below: <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{\pi }}={\frac {2{\sqrt {2}}}{9801}}\sum _{k=0}^{\infty }{\frac {(4k)!(1103+26390k)}{(k!)^{4}396^{4k}}}.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mi> π </mi> </mfrac> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn> 2 </mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn> 2 </mn> </msqrt> </mrow> </mrow> <mn> 9801 </mn> </mfrac> </mrow> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ∞ </mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false"> ( </mo> <mn> 4 </mn> <mi> k </mi> <mo stretchy="false"> ) </mo> <mo> ! </mo> <mo stretchy="false"> ( </mo> <mn> 1103 </mn> <mo> + </mo> <mn> 26390 </mn> <mi> k </mi> <mo stretchy="false"> ) </mo> </mrow> <mrow> <mo stretchy="false"> ( </mo> <mi> k </mi> <mo> ! </mo> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 4 </mn> </mrow> </msup> <msup> <mn> 396 </mn> <mrow class="MJX-TeXAtom-ORD"> <mn> 4 </mn> <mi> k </mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\frac {1}{\pi }}={\frac {2{\sqrt {2}}}{9801}}\sum _{k=0}^{\infty }{\frac {(4k)!(1103+26390k)}{(k!)^{4}396^{4k}}}.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49a59057b013e0f53abf350f5db0a6d7917eba56" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:37.518ex; height:7.176ex;" alt="{\displaystyle {\frac {1}{\pi }}={\frac {2{\sqrt {2}}}{9801}}\sum _{k=0}^{\infty }{\frac {(4k)!(1103+26390k)}{(k!)^{4}396^{4k}}}.}"> </noscript><span class="lazy-image-placeholder" style="width: 37.518ex;height: 7.176ex;vertical-align: -3.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49a59057b013e0f53abf350f5db0a6d7917eba56" data-alt="{\displaystyle {\frac {1}{\pi }}={\frac {2{\sqrt {2}}}{9801}}\sum _{k=0}^{\infty }{\frac {(4k)!(1103+26390k)}{(k!)^{4}396^{4k}}}.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> This result is based on the negative <a href="https://en-m-wikipedia-org.translate.goog/wiki/Fundamental_discriminant?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Fundamental discriminant">fundamental discriminant</a> d = −4 × 58 = −232 with class number h(d) = 2. Further, 26390 = 5 × 7 × 13 × 58 and 16 × 9801 = 3962, which is related to the fact that <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle e^{\pi {\sqrt {58}}}=396^{4}-104.000000177\dots .}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msup> <mi> e </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> π </mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn> 58 </mn> </msqrt> </mrow> </mrow> </msup> <mo> = </mo> <msup> <mn> 396 </mn> <mrow class="MJX-TeXAtom-ORD"> <mn> 4 </mn> </mrow> </msup> <mo> − </mo> <mn> 104.000000177 </mn> <mo> … </mo> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\textstyle e^{\pi {\sqrt {58}}}=396^{4}-104.000000177\dots .} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81f944da139061065021ba16d7b85c834b165a99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:34.492ex; height:3.009ex;" alt="{\textstyle e^{\pi {\sqrt {58}}}=396^{4}-104.000000177\dots .}"> </noscript><span class="lazy-image-placeholder" style="width: 34.492ex;height: 3.009ex;vertical-align: -0.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81f944da139061065021ba16d7b85c834b165a99" data-alt="{\textstyle e^{\pi {\sqrt {58}}}=396^{4}-104.000000177\dots .}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> This might be compared to <a href="https://en-m-wikipedia-org.translate.goog/wiki/Heegner_number?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Heegner number">Heegner numbers</a>, which have <a href="https://en-m-wikipedia-org.translate.goog/wiki/Ideal_class_group?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Ideal class group">class number</a> 1 and yield similar formulae. Ramanujan's series for π converges extraordinarily rapidly and forms the basis of some of the fastest algorithms used to calculate π. Truncating the sum to the first term also gives the approximation <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠9801√2/4412⁠ for π, which is correct to six decimal places; truncating it to the first two terms gives a value correct to 14 decimal places <style data-mw-deduplicate="TemplateStyles:r1033199720">.mw-parser-output div.crossreference{padding-left:0}</style> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951">(see also the more general <a href="https://en-m-wikipedia-org.translate.goog/wiki/Ramanujan%E2%80%93Sato_series?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Ramanujan–Sato series">Ramanujan–Sato series</a>). One of Ramanujan's remarkable capabilities was the rapid solution of problems, illustrated by the following anecdote about an incident in which <a href="https://en-m-wikipedia-org.translate.goog/wiki/Mahalanobis?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Mahalanobis">P. C. Mahalanobis</a> posed a problem: <style data-mw-deduplicate="TemplateStyles:r1244412712">.mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 32px}.mw-parser-output .templatequotecite{line-height:1.5em;text-align:left;margin-top:0}@media(min-width:500px){.mw-parser-output .templatequotecite{padding-left:1.6em}}</style> <blockquote class="templatequote"> Imagine that you are on a street with houses marked 1 through n. There is a house in between (x) such that the sum of the house numbers to the left of it equals the sum of the house numbers to its right. If n is between 50 and 500, what are n and x?' This is a bivariate problem with multiple solutions. Ramanujan thought about it and gave the answer with a twist: He gave a <a href="https://en-m-wikipedia-org.translate.goog/wiki/Continued_fraction?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Continued fraction">continued fraction</a>. 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."<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-118">[117]</a><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-119">[118]</a> </blockquote> His intuition also led him to derive some previously unknown <a href="https://en-m-wikipedia-org.translate.goog/wiki/Identity_(mathematics)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Identity (mathematics)">identities</a>, such as <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&\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 (}{\frac {3}{4}}{\bigr )}}{\pi }}={\frac {8\pi ^{3}}{\Gamma ^{4}{\bigl (}{\frac {1}{4}}{\bigr )}}}\end{aligned}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.9em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd></mtd> <mtd> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mn> 2 </mn> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ∞ </mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo stretchy="false"> ( </mo> <mi> n </mi> <mi> θ </mi> <mo stretchy="false"> ) </mo> </mrow> <mrow> <mi> cosh </mi> <mo> ⁡ </mo> <mo stretchy="false"> ( </mo> <mi> n </mi> <mi> π </mi> <mo stretchy="false"> ) </mo> </mrow> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mn> 2 </mn> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ∞ </mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi> cosh </mi> <mo> ⁡ </mo> <mo stretchy="false"> ( </mo> <mi> n </mi> <mi> θ </mi> <mo stretchy="false"> ) </mo> </mrow> <mrow> <mi> cosh </mi> <mo> ⁡ </mo> <mo stretchy="false"> ( </mo> <mi> n </mi> <mi> π </mi> <mo stretchy="false"> ) </mo> </mrow> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> <mn> 2 </mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn> 2 </mn> <msup> <mi mathvariant="normal"> Γ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 4 </mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em"> ( </mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 3 </mn> <mn> 4 </mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em"> ) </mo> </mrow> </mrow> </mrow> <mi> π </mi> </mfrac> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn> 8 </mn> <msup> <mi> π </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msup> </mrow> <mrow> <msup> <mi mathvariant="normal"> Γ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 4 </mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em"> ( </mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em"> ) </mo> </mrow> </mrow> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{aligned}&\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 (}{\frac {3}{4}}{\bigr )}}{\pi }}={\frac {8\pi ^{3}}{\Gamma ^{4}{\bigl (}{\frac {1}{4}}{\bigr )}}}\end{aligned}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3affe3db147ab2c00c8ea31e9620380c20c7aafb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.94ex; margin-bottom: -0.231ex; width:56.214ex; height:17.509ex;" alt="{\displaystyle {\begin{aligned}&\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 (}{\frac {3}{4}}{\bigr )}}{\pi }}={\frac {8\pi ^{3}}{\Gamma ^{4}{\bigl (}{\frac {1}{4}}{\bigr )}}}\end{aligned}}}"> </noscript><span class="lazy-image-placeholder" style="width: 56.214ex;height: 17.509ex;vertical-align: -7.94ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3affe3db147ab2c00c8ea31e9620380c20c7aafb" data-alt="{\displaystyle {\begin{aligned}&\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 (}{\frac {3}{4}}{\bigr )}}{\pi }}={\frac {8\pi ^{3}}{\Gamma ^{4}{\bigl (}{\frac {1}{4}}{\bigr )}}}\end{aligned}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> for all θ such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\Re (\theta )|<\pi }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mi mathvariant="normal"> ℜ </mi> <mo stretchy="false"> ( </mo> <mi> θ </mi> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mo> < </mo> <mi> π </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle |\Re (\theta )|<\pi } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a8d5474eacb5af80de8a543eef4dfbfd13e4e2b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.548ex; height:2.843ex;" alt="{\displaystyle |\Re (\theta )|<\pi }"> </noscript><span class="lazy-image-placeholder" style="width: 10.548ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a8d5474eacb5af80de8a543eef4dfbfd13e4e2b" data-alt="{\displaystyle |\Re (\theta )|<\pi }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\Im (\theta )|<\pi }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mi mathvariant="normal"> ℑ </mi> <mo stretchy="false"> ( </mo> <mi> θ </mi> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mo> < </mo> <mi> π </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle |\Im (\theta )|<\pi } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0641d9f7ae5a11596cb8d9d3ac78af4ec2814307" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.912ex; height:2.843ex;" alt="{\displaystyle |\Im (\theta )|<\pi }"> </noscript><span class="lazy-image-placeholder" style="width: 9.912ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0641d9f7ae5a11596cb8d9d3ac78af4ec2814307" data-alt="{\displaystyle |\Im (\theta )|<\pi }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , where Γ(z) is the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Gamma_function?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Gamma function">gamma function</a>, and related to a special value of the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Dedekind_eta_function?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Special_values" title="Dedekind eta function">Dedekind eta function</a>. Expanding into series of powers and equating coefficients of θ0, θ4, and θ8 gives some deep identities for the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Hyperbolic_secant?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Hyperbolic secant">hyperbolic secant</a>. In 1918, Hardy and Ramanujan studied the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Partition_function_(number_theory)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Partition function (number theory)">partition function</a> P(n) extensively. They gave a non-convergent asymptotic series that permits exact computation of the number of partitions of an integer. In 1937, <a href="https://en-m-wikipedia-org.translate.goog/wiki/Hans_Rademacher?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Hans Rademacher">Hans Rademacher</a> 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 <a href="https://en-m-wikipedia-org.translate.goog/wiki/Hardy%E2%80%93Littlewood_circle_method?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Hardy–Littlewood circle method">circle method</a>.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-120">[119]</a> In the last year of his life, Ramanujan discovered <a href="https://en-m-wikipedia-org.translate.goog/wiki/Mock_theta_function?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Mock theta function">mock theta functions</a>.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Fox-121">[120]</a> For many years, these functions were a mystery, but they are now known to be the holomorphic parts of harmonic weak <a href="https://en-m-wikipedia-org.translate.goog/wiki/Maass_forms?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Maass forms">Maass forms</a>. <div class="mw-heading mw-heading3"> <h3 id="The_Ramanujan_conjecture">The Ramanujan conjecture</h3> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"> <div role="note" class="hatnote navigation-not-searchable"> Main article: <a href="https://en-m-wikipedia-org.translate.goog/wiki/Ramanujan%E2%80%93Petersson_conjecture?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Ramanujan–Petersson conjecture">Ramanujan–Petersson conjecture</a> </div> 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 <a href="https://en-m-wikipedia-org.translate.goog/wiki/Andr%C3%A9_Weil?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="André Weil">André Weil</a> in algebraic geometry opened up new areas of research. That <a href="https://en-m-wikipedia-org.translate.goog/wiki/Ramanujan_conjecture?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Ramanujan conjecture">Ramanujan conjecture</a> is an assertion on the size of the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Ramanujan_tau_function?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Ramanujan tau function">tau-function</a>, which has a generating function as the discriminant modular form Δ(q), a typical <a href="https://en-m-wikipedia-org.translate.goog/wiki/Cusp_form?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Cusp form">cusp form</a> in the theory of <a href="https://en-m-wikipedia-org.translate.goog/wiki/Modular_forms?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Modular forms">modular forms</a>. It was finally proven in 1973, as a consequence of <a href="https://en-m-wikipedia-org.translate.goog/wiki/Pierre_Deligne?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Pierre Deligne">Pierre Deligne</a>'s proof of the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Weil_conjectures?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Weil conjectures">Weil conjectures</a>. The reduction step involved is complicated. Deligne won a <a href="https://en-m-wikipedia-org.translate.goog/wiki/Fields_Medal?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Fields Medal">Fields Medal</a> in 1978 for that work.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-ono-8">[7]</a><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-122">[121]</a> In his paper "On certain arithmetical functions", Ramanujan defined the so-called delta-function, whose coefficients are called τ(n) (the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Ramanujan_tau_function?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Ramanujan tau function">Ramanujan tau function</a>).<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-123">[122]</a> He proved many congruences for these numbers, such as τ(p) ≡ 1 + p11 mod 691 for primes p. This congruence (and others like it that Ramanujan proved) inspired <a href="https://en-m-wikipedia-org.translate.goog/wiki/Jean-Pierre_Serre?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Jean-Pierre Serre">Jean-Pierre Serre</a> (1954 Fields Medalist) to conjecture that there is a theory of <a href="https://en-m-wikipedia-org.translate.goog/wiki/Galois_module?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Galois module">Galois representations</a> that "explains" these congruences and more generally all modular forms. 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 <a href="https://en-m-wikipedia-org.translate.goog/wiki/Fermat%27s_Last_Theorem?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Fermat's Last Theorem">Fermat's Last Theorem</a> proceeds by first reinterpreting <a href="https://en-m-wikipedia-org.translate.goog/wiki/Elliptic_curve?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Elliptic curve">elliptic curves</a> and modular forms in terms of these Galois representations. Without this theory, there would be no proof of Fermat's Last Theorem.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-124">[123]</a> <div class="mw-heading mw-heading3"> <h3 id="Ramanujan's_notebooks">Ramanujan's notebooks</h3> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"> <div role="note" class="hatnote navigation-not-searchable"> Further information: <a href="https://en-m-wikipedia-org.translate.goog/wiki/Ramanujan%27s_lost_notebook?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Ramanujan's lost notebook">Ramanujan's lost notebook</a> </div> While still in Madras, Ramanujan recorded the bulk of his results in four notebooks of <a href="https://en-m-wikipedia-org.translate.goog/wiki/Loose_leaf?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Loose leaf">looseleaf</a> 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 <a href="https://en-m-wikipedia-org.translate.goog/wiki/Bruce_C._Berndt?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Bruce C. Berndt">Bruce C. Berndt</a>, 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 <a href="https://en-m-wikipedia-org.translate.goog/wiki/Slate_(writing)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Slate (writing)">slate</a>, after which he transferred the final results to paper. At the time, slates were commonly used by mathematics students in the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Madras_Presidency?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Madras Presidency">Madras Presidency</a>. He was also quite likely to have been influenced by the style of <a href="https://en-m-wikipedia-org.translate.goog/wiki/G._S._Carr?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="G. S. Carr">G. S. Carr</a>'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.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Bruce_Berndt_on_Ramanujan-125">[124]</a> 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 <a href="https://en-m-wikipedia-org.translate.goog/wiki/G._N._Watson?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="G. N. Watson">G. N. Watson</a>, <a href="https://en-m-wikipedia-org.translate.goog/wiki/Bertram_Martin_Wilson?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Bertram Martin Wilson">B. M. Wilson</a>, and Bruce Berndt.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Bruce_Berndt_on_Ramanujan-125">[124]</a> In 1976, <a href="https://en-m-wikipedia-org.translate.goog/wiki/George_Andrews_(mathematician)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="George Andrews (mathematician)">George Andrews</a> rediscovered a fourth notebook with 87 unorganised pages, the so-called <a href="https://en-m-wikipedia-org.translate.goog/wiki/Ramanujan%27s_lost_notebook?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Ramanujan's lost notebook">"lost notebook"</a>.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-lostnotebook-106">[105]</a> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(7)"> <h2 id="Hardy–Ramanujan_number_1729">Hardy–Ramanujan number 1729</h2> </div> <section class="mf-section-7 collapsible-block" id="mf-section-7"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"> <div role="note" class="hatnote navigation-not-searchable"> Main article: <a href="https://en-m-wikipedia-org.translate.goog/wiki/1729_(number)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="1729 (number)">1729 (number)</a> </div> 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:<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-126">[125]</a> <blockquote> I remember once going to see him when he was ill at <a href="https://en-m-wikipedia-org.translate.goog/wiki/Putney?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Putney">Putney</a>. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a <a href="https://en-m-wikipedia-org.translate.goog/wiki/Interesting_number_paradox?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Interesting number paradox">dull one</a>, 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 <a href="https://en-m-wikipedia-org.translate.goog/wiki/Sum_of_two_cubes?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Sum of two cubes">sum of two cubes</a> in two different ways." </blockquote> Immediately before this anecdote, Hardy quoted Littlewood as saying, "Every positive integer was one of [Ramanujan's] personal friends."<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-127">[126]</a> The two different ways are: <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1729=1^{3}+12^{3}=9^{3}+10^{3}.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn> 1729 </mn> <mo> = </mo> <msup> <mn> 1 </mn> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msup> <mo> + </mo> <msup> <mn> 12 </mn> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msup> <mo> = </mo> <msup> <mn> 9 </mn> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msup> <mo> + </mo> <msup> <mn> 10 </mn> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msup> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle 1729=1^{3}+12^{3}=9^{3}+10^{3}.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/030da08e579aa8b8b641d1a1c0e78d417f0465cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:28.366ex; height:2.843ex;" alt="{\displaystyle 1729=1^{3}+12^{3}=9^{3}+10^{3}.}"> </noscript><span class="lazy-image-placeholder" style="width: 28.366ex;height: 2.843ex;vertical-align: -0.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/030da08e579aa8b8b641d1a1c0e78d417f0465cf" data-alt="{\displaystyle 1729=1^{3}+12^{3}=9^{3}+10^{3}.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> Generalisations of this idea have created the notion of "<a href="https://en-m-wikipedia-org.translate.goog/wiki/Taxicab_number?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Taxicab number">taxicab numbers</a>". </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(8)"> <h2 id="Mathematicians'_views_of_Ramanujan">Mathematicians' views of Ramanujan</h2> </div> <section class="mf-section-8 collapsible-block" id="mf-section-8"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1224211176"> <div class="quotebox pullquote floatright" style="width:30%; ;"> <blockquote class="quotebox-quote left-aligned" style=""> "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." </blockquote> <cite class="left-aligned" style="">—Freeman Dyson on how another such genius might appear anywhere<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Ross_2017_u486-128">[127]</a></cite> </div> 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: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"> <blockquote class="templatequote"> 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.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Hardy1920-77">[76]</a> </blockquote> Hardy further said:<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-129">[128]</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"> <blockquote class="templatequote"> 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 <a href="https://en-m-wikipedia-org.translate.goog/wiki/Modular_equation?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Modular equation">modular equations</a> 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 <a href="https://en-m-wikipedia-org.translate.goog/wiki/Riemann_zeta_function?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Riemann zeta function">zeta function</a> 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 <a href="https://en-m-wikipedia-org.translate.goog/wiki/Doubly_periodic_function?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Doubly periodic function">doubly periodic function</a> or of <a href="https://en-m-wikipedia-org.translate.goog/wiki/Cauchy%27s_integral_theorem?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Cauchy's integral theorem">Cauchy's theorem</a>, and had indeed but the vaguest idea of what a function of a <a href="https://en-m-wikipedia-org.translate.goog/wiki/Complex_variable?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Complex variable">complex variable</a> was..." </blockquote> 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 xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{n}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle x^{n}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/150d38e238991bc4d0689ffc9d2a852547d2658d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.548ex; height:2.343ex;" alt="{\displaystyle x^{n}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.548ex;height: 2.343ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/150d38e238991bc4d0689ffc9d2a852547d2658d" data-alt="{\displaystyle x^{n}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(1-2x+2x^{4}-2x^{9}+\cdots \right)^{-1}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> − </mo> <mn> 2 </mn> <mi> x </mi> <mo> + </mo> <mn> 2 </mn> <msup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 4 </mn> </mrow> </msup> <mo> − </mo> <mn> 2 </mn> <msup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 9 </mn> </mrow> </msup> <mo> + </mo> <mo> ⋯ </mo> </mrow> <mo> ) </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> <mn> 1 </mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \left(1-2x+2x^{4}-2x^{9}+\cdots \right)^{-1}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ab3512523eda5a5dd672595ccab4ed6f64ee3eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:29.295ex; height:3.843ex;" alt="{\displaystyle \left(1-2x+2x^{4}-2x^{9}+\cdots \right)^{-1}}"> </noscript><span class="lazy-image-placeholder" style="width: 29.295ex;height: 3.843ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ab3512523eda5a5dd672595ccab4ed6f64ee3eb" data-alt="{\displaystyle \left(1-2x+2x^{4}-2x^{9}+\cdots \right)^{-1}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  is the integer nearest to <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{4n}}\left(\cosh(\pi {\sqrt {n}})-{\frac {\sinh(\pi {\sqrt {n}})}{\pi {\sqrt {n}}}}\right).}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mrow> <mn> 4 </mn> <mi> n </mi> </mrow> </mfrac> </mrow> <mrow> <mo> ( </mo> <mrow> <mi> cosh </mi> <mo> ⁡ </mo> <mo stretchy="false"> ( </mo> <mi> π </mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi> n </mi> </msqrt> </mrow> <mo stretchy="false"> ) </mo> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi> sinh </mi> <mo> ⁡ </mo> <mo stretchy="false"> ( </mo> <mi> π </mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi> n </mi> </msqrt> </mrow> <mo stretchy="false"> ) </mo> </mrow> <mrow> <mi> π </mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi> n </mi> </msqrt> </mrow> </mrow> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\frac {1}{4n}}\left(\cosh(\pi {\sqrt {n}})-{\frac {\sinh(\pi {\sqrt {n}})}{\pi {\sqrt {n}}}}\right).} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31d6c3b0bc4bd50fe1fac8d2ce7fdd39f5ce1db3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:33.408ex; height:6.843ex;" alt="{\displaystyle {\frac {1}{4n}}\left(\cosh(\pi {\sqrt {n}})-{\frac {\sinh(\pi {\sqrt {n}})}{\pi {\sqrt {n}}}}\right).}"> </noscript><span class="lazy-image-placeholder" style="width: 33.408ex;height: 6.843ex;vertical-align: -2.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31d6c3b0bc4bd50fe1fac8d2ce7fdd39f5ce1db3" data-alt="{\displaystyle {\frac {1}{4n}}\left(\cosh(\pi {\sqrt {n}})-{\frac {\sinh(\pi {\sqrt {n}})}{\pi {\sqrt {n}}}}\right).}" data-class="mwe-math-fallback-image-display mw-invert skin-invert"> This "was one of the most fruitful he ever made, since it ended by leading us to all our joint work on partitions".<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-:0-130">[129]</a> 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."<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-cola1-131">[130]</a> He also said that he had "never met his equal, and can compare him only with Euler or Jacobi".<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-cola1-131">[130]</a> 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.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-:0-130">[129]</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"> <blockquote class="templatequote"> 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.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-:0-130">[129]</a> </blockquote> 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 <a href="https://en-m-wikipedia-org.translate.goog/wiki/Prime_number_theorem?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Prime number theorem">prime number theorem</a>. In analysis, he worked on the elliptic functions and the analytic theory of numbers. In <a href="https://en-m-wikipedia-org.translate.goog/wiki/Analytic_number_theory?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Analytic number theory">analytic number theory</a>, 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 <a href="https://en-m-wikipedia-org.translate.goog/wiki/Analytic_function?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Analytic function">analytic functions</a>".<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-:0-130">[129]</a> 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.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-alyangar1995-132">[131]</a> K. Srinivasa Rao has said,<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-133">[132]</a> "As for his place in the world of Mathematics, we quote Bruce C. Berndt: '<a href="https://en-m-wikipedia-org.translate.goog/wiki/Paul_Erd%C5%91s?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Paul Erdős">Paul Erdős</a> 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, <a href="https://en-m-wikipedia-org.translate.goog/wiki/David_Hilbert?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="David Hilbert">David Hilbert</a> 80 and Ramanujan 100.'" During a May 2011 lecture at <a href="https://en-m-wikipedia-org.translate.goog/wiki/IIT_Madras?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="IIT Madras">IIT Madras</a>, 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.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Fox-121">[120]</a><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-134">[133]</a> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(9)"> <h2 id="Posthumous_recognition">Posthumous recognition</h2> </div> <section class="mf-section-9 collapsible-block" id="mf-section-9"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"> <div role="note" class="hatnote navigation-not-searchable"> Further information: <a href="https://en-m-wikipedia-org.translate.goog/wiki/List_of_things_named_after_Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="List of things named after Srinivasa Ramanujan">List of things named after Srinivasa Ramanujan</a> </div> <figure class="mw-default-size" typeof="mw:File/Thumb"> <a href="https://en-m-wikipedia-org.translate.goog/wiki/File:Srinivasa_Ramanujam_bust_BITM.JPG?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-file-description"> <noscript> <img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d3/Srinivasa_Ramanujam_bust_BITM.JPG/170px-Srinivasa_Ramanujam_bust_BITM.JPG" decoding="async" width="170" height="227" class="mw-file-element" data-file-width="2736" data-file-height="3648"> </noscript><span class="lazy-image-placeholder" style="width: 170px;height: 227px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d3/Srinivasa_Ramanujam_bust_BITM.JPG/170px-Srinivasa_Ramanujam_bust_BITM.JPG" data-width="170" data-height="227" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d3/Srinivasa_Ramanujam_bust_BITM.JPG/255px-Srinivasa_Ramanujam_bust_BITM.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d3/Srinivasa_Ramanujam_bust_BITM.JPG/340px-Srinivasa_Ramanujam_bust_BITM.JPG 2x" data-class="mw-file-element"> </a> <figcaption> Bust of Ramanujan in the garden of <a href="https://en-m-wikipedia-org.translate.goog/wiki/Birla_Industrial_%26_Technological_Museum?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Birla Industrial & Technological Museum">Birla Industrial & Technological Museum</a> in <a href="https://en-m-wikipedia-org.translate.goog/wiki/Kolkata?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Kolkata">Kolkata</a>, India </figcaption> </figure> The year after his death, <a href="https://en-m-wikipedia-org.translate.goog/wiki/Nature_(journal)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Nature (journal)">Nature</a> listed Ramanujan among other distinguished scientists and mathematicians on a "Calendar of Scientific Pioneers" who had achieved eminence.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-135">[134]</a> Ramanujan's home state of <a href="https://en-m-wikipedia-org.translate.goog/wiki/Tamil_Nadu?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Tamil Nadu">Tamil Nadu</a> celebrates 22 December (Ramanujan's birthday) as 'State IT Day'. Stamps picturing Ramanujan were issued by the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Government_of_India?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Government of India">government of India</a> in 1962, 2011, 2012 and 2016.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-136">[135]</a> Since Ramanujan's centennial year, his birthday, 22 December, has been annually celebrated as Ramanujan Day by the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Government_Arts_College,_Kumbakonam?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Government Arts College, Kumbakonam">Government Arts College, Kumbakonam</a>, where he studied, and at the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Indian_Institute_of_Technology_Madras?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Indian Institute of Technology Madras">IIT Madras</a> in <a href="https://en-m-wikipedia-org.translate.goog/wiki/Chennai?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Chennai">Chennai</a>. The <a href="https://en-m-wikipedia-org.translate.goog/wiki/International_Centre_for_Theoretical_Physics?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="International Centre for Theoretical Physics">International Centre for Theoretical Physics</a> (ICTP) has created a prize in Ramanujan's name for young mathematicians from developing countries in cooperation with the <a href="https://en-m-wikipedia-org.translate.goog/wiki/International_Mathematical_Union?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="International Mathematical Union">International Mathematical Union</a>, which nominates members of the prize committee. <a href="https://en-m-wikipedia-org.translate.goog/wiki/Shanmugha_Arts,_Science,_Technology_%26_Research_Academy?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Shanmugha Arts, Science, Technology & Research Academy">SASTRA University</a>, a private university based in <a href="https://en-m-wikipedia-org.translate.goog/wiki/Tamil_Nadu?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Tamil Nadu">Tamil Nadu</a>, has instituted the <a href="https://en-m-wikipedia-org.translate.goog/wiki/SASTRA_Ramanujan_Prize?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="SASTRA Ramanujan Prize">SASTRA Ramanujan Prize</a> of <a href="https://en-m-wikipedia-org.translate.goog/wiki/US%24?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="US$">US$</a>10,000 to be given annually to a mathematician not exceeding age 32 for outstanding contributions in an area of mathematics influenced by Ramanujan.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-src-137">[136]</a> 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.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-src-137">[136]</a> 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.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-138">[137]</a> Then Indian Prime Minister <a href="https://en-m-wikipedia-org.translate.goog/wiki/Manmohan_Singh?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Manmohan Singh">Manmohan Singh</a> also declared that 2012 would be celebrated as <a href="https://en-m-wikipedia-org.translate.goog/wiki/National_Mathematics_Year?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="National Mathematics Year">National Mathematics Year</a> and 22 December as <a href="https://en-m-wikipedia-org.translate.goog/wiki/National_Mathematics_Day_(India)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="National Mathematics Day (India)">National Mathematics Day</a> of India.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-139">[138]</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Ramanujan_IT_City?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Ramanujan IT City">Ramanujan IT City</a> is an information technology (IT) <a href="https://en-m-wikipedia-org.translate.goog/wiki/Special_economic_zone?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Special economic zone">special economic zone</a> (SEZ) in <a href="https://en-m-wikipedia-org.translate.goog/wiki/Chennai?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Chennai">Chennai</a> that was built in 2011. Situated next to the <a href="https://en-m-wikipedia-org.translate.goog/wiki/TIDEL_Park?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="TIDEL Park">Tidel Park</a>, it includes 25 acres (10 ha) with two zones, with a total area of 5.7 million square feet (530,000 m2), including 4.5 million square feet (420,000 m2) of office space.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-140">[139]</a> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(10)"> <h2 id="Commemorative_postal_stamps">Commemorative postal stamps</h2> </div> <section class="mf-section-10 collapsible-block" id="mf-section-10"> Commemorative stamps released by <a href="https://en-m-wikipedia-org.translate.goog/wiki/India_Post?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="India Post">India Post</a> (by year): <style data-mw-deduplicate="TemplateStyles:r1237032888/mw-parser-output/.tmulti">.mw-parser-output .tmulti .multiimageinner{display:flex;flex-direction:column}.mw-parser-output .tmulti .trow{display:flex;flex-direction:row;clear:left;flex-wrap:wrap;width:100%;box-sizing:border-box}.mw-parser-output .tmulti .tsingle{margin:1px;float:left}.mw-parser-output .tmulti .theader{clear:both;font-weight:bold;text-align:center;align-self:center;background-color:transparent;width:100%}.mw-parser-output .tmulti .thumbcaption{background-color:transparent}.mw-parser-output .tmulti .text-align-left{text-align:left}.mw-parser-output .tmulti .text-align-right{text-align:right}.mw-parser-output .tmulti .text-align-center{text-align:center}@media all and (max-width:720px){.mw-parser-output .tmulti .thumbinner{width:100%!important;box-sizing:border-box;max-width:none!important;align-items:center}.mw-parser-output .tmulti .trow{justify-content:center}.mw-parser-output .tmulti .tsingle{float:none!important;max-width:100%!important;box-sizing:border-box;text-align:center}.mw-parser-output .tmulti .tsingle .thumbcaption{text-align:left}.mw-parser-output .tmulti .trow>.thumbcaption{text-align:center}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .tmulti .multiimageinner img{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .tmulti .multiimageinner img{background-color:white}}</style> <div class="thumb tmulti tnone center"> <div class="thumbinner multiimageinner" style="width:856px;max-width:856px"> <div class="trow"> <div class="tsingle" style="width:202px;max-width:202px"> <div class="thumbimage"> <a href="https://en-m-wikipedia-org.translate.goog/wiki/File:Srinivasa_Ramanujan_1962_stamp_of_India.jpg?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-file-description"> <noscript> <img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/90/Srinivasa_Ramanujan_1962_stamp_of_India.jpg/200px-Srinivasa_Ramanujan_1962_stamp_of_India.jpg" decoding="async" width="200" height="171" class="mw-file-element" data-file-width="1024" data-file-height="876"> </noscript><span class="lazy-image-placeholder" style="width: 200px;height: 171px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/9/90/Srinivasa_Ramanujan_1962_stamp_of_India.jpg/200px-Srinivasa_Ramanujan_1962_stamp_of_India.jpg" data-alt="" data-width="200" data-height="171" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/90/Srinivasa_Ramanujan_1962_stamp_of_India.jpg/300px-Srinivasa_Ramanujan_1962_stamp_of_India.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/90/Srinivasa_Ramanujan_1962_stamp_of_India.jpg/400px-Srinivasa_Ramanujan_1962_stamp_of_India.jpg 2x" data-class="mw-file-element"> </a> </div> <div class="thumbcaption"> 1962 </div> </div> <div class="tsingle" style="width:222px;max-width:222px"> <div class="thumbimage"> <a href="https://en-m-wikipedia-org.translate.goog/wiki/File:Srinivasa_Ramanujan_2011_stamp_of_India.jpg?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-file-description"> <noscript> <img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8e/Srinivasa_Ramanujan_2011_stamp_of_India.jpg/220px-Srinivasa_Ramanujan_2011_stamp_of_India.jpg" decoding="async" width="220" height="166" class="mw-file-element" data-file-width="923" data-file-height="695"> </noscript><span class="lazy-image-placeholder" style="width: 220px;height: 166px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8e/Srinivasa_Ramanujan_2011_stamp_of_India.jpg/220px-Srinivasa_Ramanujan_2011_stamp_of_India.jpg" data-alt="" data-width="220" data-height="166" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8e/Srinivasa_Ramanujan_2011_stamp_of_India.jpg/330px-Srinivasa_Ramanujan_2011_stamp_of_India.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8e/Srinivasa_Ramanujan_2011_stamp_of_India.jpg/440px-Srinivasa_Ramanujan_2011_stamp_of_India.jpg 2x" data-class="mw-file-element"> </a> </div> <div class="thumbcaption"> 2011 </div> </div> <div class="tsingle" style="width:272px;max-width:272px"> <div class="thumbimage"> <a href="https://en-m-wikipedia-org.translate.goog/wiki/File:Srinivasa_Ramanujan_2012_stamp_of_India.jpg?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-file-description"> <noscript> <img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Srinivasa_Ramanujan_2012_stamp_of_India.jpg/270px-Srinivasa_Ramanujan_2012_stamp_of_India.jpg" decoding="async" width="270" height="163" class="mw-file-element" data-file-width="727" data-file-height="440"> </noscript><span class="lazy-image-placeholder" style="width: 270px;height: 163px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Srinivasa_Ramanujan_2012_stamp_of_India.jpg/270px-Srinivasa_Ramanujan_2012_stamp_of_India.jpg" data-alt="" data-width="270" data-height="163" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Srinivasa_Ramanujan_2012_stamp_of_India.jpg/405px-Srinivasa_Ramanujan_2012_stamp_of_India.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Srinivasa_Ramanujan_2012_stamp_of_India.jpg/540px-Srinivasa_Ramanujan_2012_stamp_of_India.jpg 2x" data-class="mw-file-element"> </a> </div> <div class="thumbcaption"> 2012 </div> </div> <div class="tsingle" style="width:152px;max-width:152px"> <div class="thumbimage"> <a href="https://en-m-wikipedia-org.translate.goog/wiki/File:Srinivasa_Ramanujan_2016_stamp_of_India.jpg?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-file-description"> <noscript> <img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d4/Srinivasa_Ramanujan_2016_stamp_of_India.jpg/150px-Srinivasa_Ramanujan_2016_stamp_of_India.jpg" decoding="async" width="150" height="168" class="mw-file-element" data-file-width="239" data-file-height="268"> </noscript><span class="lazy-image-placeholder" style="width: 150px;height: 168px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d4/Srinivasa_Ramanujan_2016_stamp_of_India.jpg/150px-Srinivasa_Ramanujan_2016_stamp_of_India.jpg" data-alt="" data-width="150" data-height="168" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d4/Srinivasa_Ramanujan_2016_stamp_of_India.jpg/225px-Srinivasa_Ramanujan_2016_stamp_of_India.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/d/d4/Srinivasa_Ramanujan_2016_stamp_of_India.jpg 2x" data-class="mw-file-element"> </a> </div> <div class="thumbcaption"> 2016 </div> </div> </div> </div> </div> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(11)"> <h2 id="In_popular_culture">In popular culture</h2> </div> <section class="mf-section-11 collapsible-block" id="mf-section-11"> <ul> <li>The Man Who Loved Numbers is a 1988 PBS NOVA documentary about Ramanujan (S15, E9).<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-141">[140]</a></li> <li><a href="https://en-m-wikipedia-org.translate.goog/wiki/The_Man_Who_Knew_Infinity?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="The Man Who Knew Infinity">The Man Who Knew Infinity</a> is a 2015 film based on <a href="https://en-m-wikipedia-org.translate.goog/wiki/Robert_Kanigel?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Robert Kanigel">Kanigel's</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/The_Man_Who_Knew_Infinity_(book)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="The Man Who Knew Infinity (book)">book of the same name</a>. British actor <a href="https://en-m-wikipedia-org.translate.goog/wiki/Dev_Patel?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Dev Patel">Dev Patel</a> portrays Ramanujan.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-142">[141]</a><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-143">[142]</a><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-144">[143]</a></li> <li><a href="https://en-m-wikipedia-org.translate.goog/wiki/Ramanujan_(film)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Ramanujan (film)">Ramanujan</a>, an Indo-British collaboration film chronicling Ramanujan's life, was released in 2014 by the independent film company <a href="https://en-m-wikipedia-org.translate.goog/wiki/Ramanujan_(film)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Camphor_Cinema" title="Ramanujan (film)">Camphor Cinema</a>.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-145">[144]</a> The cast and crew include director <a href="https://en-m-wikipedia-org.translate.goog/wiki/Gnana_Rajasekaran?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Gnana Rajasekaran">Gnana Rajasekaran</a>, cinematographer <a href="https://en-m-wikipedia-org.translate.goog/wiki/Sunny_Joseph?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Sunny Joseph">Sunny Joseph</a> and editor <a href="https://en-m-wikipedia-org.translate.goog/wiki/B._Lenin?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="B. Lenin">B. Lenin</a>.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-146">[145]</a><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-147">[146]</a> Indian and English stars <a href="https://en-m-wikipedia-org.translate.goog/wiki/Abhinay_Vaddi?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Abhinay Vaddi">Abhinay Vaddi</a>, <a href="https://en-m-wikipedia-org.translate.goog/wiki/Suhasini_Maniratnam?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Suhasini Maniratnam">Suhasini Maniratnam</a>, <a href="https://en-m-wikipedia-org.translate.goog/wiki/Bhama_Kurup?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Bhama Kurup">Bhama</a>, Kevin McGowan and <a href="https://en-m-wikipedia-org.translate.goog/wiki/Michael_Lieber?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Michael Lieber">Michael Lieber</a> star in pivotal roles.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-148">[147]</a></li> <li>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.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-149">[148]</a></li> <li>Ramanujan (The Man Who Reshaped 20th Century Mathematics), an Indian docudrama film directed by Akashdeep released in 2018.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-150">[149]</a></li> <li>M. N. Krish's thriller novel The Steradian Trail weaves Ramanujan and his accidental discovery into its plot connecting religion, mathematics, finance and economics.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-151">[150]</a><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-152">[151]</a></li> <li>Partition, a play by Ira Hauptman about Hardy and Ramanujan, was first performed in 2013.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-153">[152]</a><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-154">[153]</a><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-155">[154]</a><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-156">[155]</a></li> <li>The play First Class Man by Alter Ego Productions<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-157">[156]</a> 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 <a href="https://en-m-wikipedia-org.translate.goog/wiki/Roger_Spottiswoode?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Roger Spottiswoode">Roger Spottiswoode</a>, best known for his <a href="https://en-m-wikipedia-org.translate.goog/wiki/James_Bond_film?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="James Bond film">James Bond film</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Tomorrow_Never_Dies?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Tomorrow Never Dies">Tomorrow Never Dies</a>, is working on the film version, starring <a href="https://en-m-wikipedia-org.translate.goog/wiki/Siddharth_(actor)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Siddharth (actor)">Siddharth</a>.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-158">[157]</a></li> <li><a href="https://en-m-wikipedia-org.translate.goog/wiki/A_Disappearing_Number?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="A Disappearing Number">A Disappearing Number</a> is a British stage production by the company <a href="https://en-m-wikipedia-org.translate.goog/wiki/Complicite?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Complicite">Complicite</a> that explores the relationship between Hardy and Ramanujan.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-159">[158]</a></li> <li><a href="https://en-m-wikipedia-org.translate.goog/wiki/David_Leavitt?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="David Leavitt">David Leavitt</a>'s novel <a href="https://en-m-wikipedia-org.translate.goog/wiki/The_Indian_Clerk?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="The Indian Clerk">The Indian Clerk</a> explores the events following Ramanujan's letter to Hardy.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-160">[159]</a><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-161">[160]</a></li> <li><a href="https://en-m-wikipedia-org.translate.goog/wiki/Google?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Google">Google</a> honoured Ramanujan on his 125th birth anniversary by replacing its logo with a <a href="https://en-m-wikipedia-org.translate.goog/wiki/Google_Doodles?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Google Doodles">doodle</a> on its home page.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-162">[161]</a><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-163">[162]</a></li> <li>Ramanujan was mentioned in the 1997 film <a href="https://en-m-wikipedia-org.translate.goog/wiki/Good_Will_Hunting?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Good Will Hunting">Good Will Hunting</a>, in a scene where professor Gerald Lambeau (<a href="https://en-m-wikipedia-org.translate.goog/wiki/Stellan_Skarsg%C3%A5rd?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Stellan Skarsgård">Stellan Skarsgård</a>) explains to Sean Maguire (<a href="https://en-m-wikipedia-org.translate.goog/wiki/Robin_Williams?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Robin Williams">Robin Williams</a>) the genius of Will Hunting (<a href="https://en-m-wikipedia-org.translate.goog/wiki/Matt_Damon?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Matt Damon">Matt Damon</a>) by comparing him to Ramanujan.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-164">[163]</a></li> </ul> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(12)"> <h2 id="Selected_papers">Selected papers</h2> </div> <section class="mf-section-12 collapsible-block" id="mf-section-12"> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style> <div class="div-col" style="column-width: 35em;"> <ul> <li><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFRamanujan1914" class="citation journal cs1">Ramanujan, S. (1914). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://archive.org/details/messengerofmathe44cambuoft/page/10/mode/2up">"Some definite integrals"</a>. Messenger Math. 44: 10–18.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Messenger+Math.&rft.atitle=Some+definite+integrals&rft.volume=44&rft.pages=10-18&rft.date=1914&rft.aulast=Ramanujan&rft.aufirst=S.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fmessengerofmathe44cambuoft%2Fpage%2F10%2Fmode%2F2up&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASrinivasa+Ramanujan" class="Z3988"></li> <li> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRamanujan1914" class="citation journal cs1">Ramanujan, S. (1914). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://archive.org/details/messengerofmathe44cambuoft/page/10/mode/2up">"Some definite integrals connected with Gauss's sums"</a>. Messenger Math. 44: 75–85.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Messenger+Math.&rft.atitle=Some+definite+integrals+connected+with+Gauss%27s+sums&rft.volume=44&rft.pages=75-85&rft.date=1914&rft.aulast=Ramanujan&rft.aufirst=S.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fmessengerofmathe44cambuoft%2Fpage%2F10%2Fmode%2F2up&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASrinivasa+Ramanujan" class="Z3988"></li> <li> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRamanujan1915" class="citation journal cs1">Ramanujan, S. (1915). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://archive.org/details/messengerofmathe45cambuoft/page/10/mode/2up">"On certain infinite series"</a>. Messenger Math. 45: 11–15.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Messenger+Math.&rft.atitle=On+certain+infinite+series&rft.volume=45&rft.pages=11-15&rft.date=1915&rft.aulast=Ramanujan&rft.aufirst=S.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fmessengerofmathe45cambuoft%2Fpage%2F10%2Fmode%2F2up&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASrinivasa+Ramanujan" class="Z3988"></li> <li> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRamanujan1915" class="citation journal cs1">Ramanujan, S. (1915). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://zenodo.org/record/1433496">"Highly Composite Numbers"</a>. Proceedings of the London Mathematical Society. 14 (1): 347–409. <a href="https://en-m-wikipedia-org.translate.goog/wiki/Doi_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://doi.org/10.1112%252Fplms%252Fs2_14.1.347">10.1112/plms/s2_14.1.347</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+London+Mathematical+Society&rft.atitle=Highly+Composite+Numbers&rft.volume=14&rft.issue=1&rft.pages=347-409&rft.date=1915&rft_id=info%3Adoi%2F10.1112%2Fplms%2Fs2_14.1.347&rft.aulast=Ramanujan&rft.aufirst=S.&rft_id=https%3A%2F%2Fzenodo.org%2Frecord%2F1433496&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASrinivasa+Ramanujan" class="Z3988"></li> <li> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRamanujan1915" class="citation journal cs1">Ramanujan, S. (1915). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://archive.org/details/dli.ernet.504289/page/131/mode/2up">"On the number of divisors of a number"</a>. The Journal of the Indian Mathematical Society. 7 (4): 131–133.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Journal+of+the+Indian+Mathematical+Society&rft.atitle=On+the+number+of+divisors+of+a+number&rft.volume=7&rft.issue=4&rft.pages=131-133&rft.date=1915&rft.aulast=Ramanujan&rft.aufirst=S.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fdli.ernet.504289%2Fpage%2F131%2Fmode%2F2up&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASrinivasa+Ramanujan" class="Z3988"></li> <li> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRamanujan1915" class="citation journal cs1">Ramanujan, S. (1915). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://archive.org/details/dli.ernet.504289/page/173/mode/2up">"Short Note: On the sum of the square roots of the first n natural numbers"</a>. The Journal of the Indian Mathematical Society. 7 (5): 173–175.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Journal+of+the+Indian+Mathematical+Society&rft.atitle=Short+Note%3A+On+the+sum+of+the+square+roots+of+the+first+n+natural+numbers&rft.volume=7&rft.issue=5&rft.pages=173-175&rft.date=1915&rft.aulast=Ramanujan&rft.aufirst=S.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fdli.ernet.504289%2Fpage%2F173%2Fmode%2F2up&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASrinivasa+Ramanujan" class="Z3988"></li> <li> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRamanujan1916" class="citation journal cs1">Ramanujan, S. (1916). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://archive.org/details/messengerofmathe45cambuoft/page/80/mode/2up">"Some formulae in the analytical theory of numbers"</a>. Messenger Math. 45: 81–84.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Messenger+Math.&rft.atitle=Some+formulae+in+the+analytical+theory+of+numbers&rft.volume=45&rft.pages=81-84&rft.date=1916&rft.aulast=Ramanujan&rft.aufirst=S.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fmessengerofmathe45cambuoft%2Fpage%2F80%2Fmode%2F2up&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASrinivasa+Ramanujan" class="Z3988"></li> <li> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRamanujan1916" class="citation journal cs1">Ramanujan, S. (1916). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://archive.org/details/messengerofmathe46cambuoft/page/72/mode/2up">"A Series for Euler's Constant γ"</a>. Messenger Math. 46: 73–80.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Messenger+Math.&rft.atitle=A+Series+for+Euler%27s+Constant+%CE%B3&rft.volume=46&rft.pages=73-80&rft.date=1916&rft.aulast=Ramanujan&rft.aufirst=S.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fmessengerofmathe46cambuoft%2Fpage%2F72%2Fmode%2F2up&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASrinivasa+Ramanujan" class="Z3988"></li> <li> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRamanujan1917" class="citation journal cs1">Ramanujan, S. (1917). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://archive.org/details/proceedingsofcam1920191721camb/page/n23/mode/2up">"On the expression of numbers in the form ax2 + by2 + cz2 + du2"</a>. Mathematical Proceedings of the Cambridge Philosophical Society. 19: 11–21.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematical+Proceedings+of+the+Cambridge+Philosophical+Society&rft.atitle=On+the+expression+of+numbers+in+the+form+%3Cspan+class%3D%22texhtml+%22+%3Eax%3Csup%3E2%3C%2Fsup%3E+%2B+by%3Csup%3E2%3C%2Fsup%3E+%2B+cz%3Csup%3E2%3C%2Fsup%3E+%2B+du%3Csup%3E2%3C%2Fsup%3E%3C%2Fspan%3E&rft.volume=19&rft.pages=11-21&rft.date=1917&rft.aulast=Ramanujan&rft.aufirst=S.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fproceedingsofcam1920191721camb%2Fpage%2Fn23%2Fmode%2F2up&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASrinivasa+Ramanujan" class="Z3988"></li> <li> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHardyRamanujan1917" class="citation journal cs1">Hardy, G. H.; Ramanujan, S. (1917). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://zenodo.org/record/1553638">"Asymptotic Formulae for the Distribution of Integers of Various Types"</a>. Proceedings of the London Mathematical Society. 16 (1): 112–132. <a href="https://en-m-wikipedia-org.translate.goog/wiki/Doi_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://doi.org/10.1112%252Fplms%252Fs2-16.1.112">10.1112/plms/s2-16.1.112</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+London+Mathematical+Society&rft.atitle=Asymptotic+Formulae+for+the+Distribution+of+Integers+of+Various+Types&rft.volume=16&rft.issue=1&rft.pages=112-132&rft.date=1917&rft_id=info%3Adoi%2F10.1112%2Fplms%2Fs2-16.1.112&rft.aulast=Hardy&rft.aufirst=G.+H.&rft.au=Ramanujan%2C+S.&rft_id=https%3A%2F%2Fzenodo.org%2Frecord%2F1553638&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASrinivasa+Ramanujan" class="Z3988"></li> <li> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHardyRamanujan1918" class="citation journal cs1"><a href="https://en-m-wikipedia-org.translate.goog/wiki/G._H._Hardy?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="G. H. Hardy">Hardy, G. H.</a>; Ramanujan, Srinivasa (1918). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://zenodo.org/record/1560148">"Asymptotic Formulae in Combinatory Analysis"</a>. Proceedings of the London Mathematical Society. 17 (1): 75–115. <a href="https://en-m-wikipedia-org.translate.goog/wiki/Doi_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://doi.org/10.1112%252Fplms%252Fs2-17.1.75">10.1112/plms/s2-17.1.75</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+London+Mathematical+Society&rft.atitle=Asymptotic+Formulae+in+Combinatory+Analysis&rft.volume=17&rft.issue=1&rft.pages=75-115&rft.date=1918&rft_id=info%3Adoi%2F10.1112%2Fplms%2Fs2-17.1.75&rft.aulast=Hardy&rft.aufirst=G.+H.&rft.au=Ramanujan%2C+Srinivasa&rft_id=https%3A%2F%2Fzenodo.org%2Frecord%2F1560148&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASrinivasa+Ramanujan" class="Z3988"></li> <li> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHardyRamanujan1918" class="citation journal cs1">Hardy, G. H.; Ramanujan, Srinivasa (1918). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://doi.org/10.1098%252Frspa.1918.0056">"On the coefficients in the expansions of certain modular functions"</a>. Proc. R. Soc. A. 95 (667): 144–155. <a href="https://en-m-wikipedia-org.translate.goog/wiki/Bibcode_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ui.adsabs.harvard.edu/abs/1918RSPSA..95..144H">1918RSPSA..95..144H</a>. <a href="https://en-m-wikipedia-org.translate.goog/wiki/Doi_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://doi.org/10.1098%252Frspa.1918.0056">10.1098/rspa.1918.0056</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proc.+R.+Soc.+A&rft.atitle=On+the+coefficients+in+the+expansions+of+certain+modular+functions&rft.volume=95&rft.issue=667&rft.pages=144-155&rft.date=1918&rft_id=info%3Adoi%2F10.1098%2Frspa.1918.0056&rft_id=info%3Abibcode%2F1918RSPSA..95..144H&rft.aulast=Hardy&rft.aufirst=G.+H.&rft.au=Ramanujan%2C+Srinivasa&rft_id=https%3A%2F%2Fdoi.org%2F10.1098%252Frspa.1918.0056&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASrinivasa+Ramanujan" class="Z3988"></li> <li> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRamanujan1919" class="citation journal cs1">Ramanujan, Srinivasa (1919). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://archive.org/details/in.ernet.dli.2015.509608/page/n89/mode/2up">"Some definite integrals"</a>. The Journal of the Indian Mathematical Society. 11 (2): 81–88.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Journal+of+the+Indian+Mathematical+Society&rft.atitle=Some+definite+integrals&rft.volume=11&rft.issue=2&rft.pages=81-88&rft.date=1919&rft.aulast=Ramanujan&rft.aufirst=Srinivasa&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fin.ernet.dli.2015.509608%2Fpage%2Fn89%2Fmode%2F2up&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASrinivasa+Ramanujan" class="Z3988"></li> <li> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRamanujan1919" class="citation journal cs1">Ramanujan, S. (1919). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://archive.org/details/in.ernet.dli.2015.509608/page/n195/mode/2up">"A proof of Bertrand's postulate"</a>. The Journal of the Indian Mathematical Society. 11 (5): 181–183.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Journal+of+the+Indian+Mathematical+Society&rft.atitle=A+proof+of+Bertrand%27s+postulate&rft.volume=11&rft.issue=5&rft.pages=181-183&rft.date=1919&rft.aulast=Ramanujan&rft.aufirst=S.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fin.ernet.dli.2015.509608%2Fpage%2Fn195%2Fmode%2F2up&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASrinivasa+Ramanujan" class="Z3988"></li> <li> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRamanujan1920" class="citation journal cs1">Ramanujan, S. (1920). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://hdl.handle.net/2027/uc1.%2524b417568?urlappend%3D%253Bseq%3D306%253Bownerid%3D9007199275659670-310">"A class of definite integrals"</a>. Quart. J. Pure. Appl. Math. 48: 294–309. <a href="https://en-m-wikipedia-org.translate.goog/wiki/Hdl_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://hdl.handle.net/2027%252Fuc1.%2524b417568?urlappend%3D%253Bseq%3D306">2027/uc1.$b417568</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Quart.+J.+Pure.+Appl.+Math.&rft.atitle=A+class+of+definite+integrals&rft.volume=48&rft.pages=294-309&rft.date=1920&rft_id=info%3Ahdl%2F2027%2Fuc1.%24b417568%3Furlappend%3D%253Bseq%3D306&rft.aulast=Ramanujan&rft.aufirst=S.&rft_id=https%3A%2F%2Fhdl.handle.net%2F2027%2Fuc1.%24b417568%3Furlappend%3D%253Bseq%3D306%253Bownerid%3D9007199275659670-310&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASrinivasa+Ramanujan" class="Z3988"></li> <li> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRamanujan1921" class="citation journal cs1">Ramanujan, S. (1921). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://zenodo.org/record/1447425">"Congruence properties of partitions"</a>. Math. Z. 9 (1–2): 147–153. <a href="https://en-m-wikipedia-org.translate.goog/wiki/Doi_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://doi.org/10.1007%252FBF01378341">10.1007/BF01378341</a>. <a href="https://en-m-wikipedia-org.translate.goog/wiki/S2CID_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://api.semanticscholar.org/CorpusID:121753215">121753215</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Math.+Z.&rft.atitle=Congruence+properties+of+partitions&rft.volume=9&rft.issue=1%E2%80%932&rft.pages=147-153&rft.date=1921&rft_id=info%3Adoi%2F10.1007%2FBF01378341&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A121753215%23id-name%3DS2CID&rft.aulast=Ramanujan&rft.aufirst=S.&rft_id=https%3A%2F%2Fzenodo.org%2Frecord%2F1447425&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASrinivasa+Ramanujan" class="Z3988"> Posthumously published extract of a longer, unpublished manuscript.</li> </ul> </div> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(13)"> <h2 id="Further_works_of_Ramanujan's_mathematics">Further works of Ramanujan's mathematics</h2> </div> <section class="mf-section-13 collapsible-block" id="mf-section-13"> <ul> <li><a href="https://en-m-wikipedia-org.translate.goog/wiki/George_E._Andrews?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="George E. Andrews">George E. Andrews</a> and <a href="https://en-m-wikipedia-org.translate.goog/wiki/Bruce_C._Berndt?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Bruce C. Berndt">Bruce C. Berndt</a>, Ramanujan's Lost Notebook: Part I (Springer, 2005, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="https://en-m-wikipedia-org.translate.goog/wiki/ISBN_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Special:BookSources/0-387-25529-X?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Special:BookSources/0-387-25529-X">0-387-25529-X</a>)<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-165">[164]</a></li> <li>George E. Andrews and Bruce C. Berndt, Ramanujan's Lost Notebook: Part II, (Springer, 2008, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="https://en-m-wikipedia-org.translate.goog/wiki/ISBN_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Special:BookSources/978-0-387-77765-8?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Special:BookSources/978-0-387-77765-8">978-0-387-77765-8</a>)</li> <li>George E. Andrews and Bruce C. Berndt, Ramanujan's Lost Notebook: Part III, (Springer, 2012, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="https://en-m-wikipedia-org.translate.goog/wiki/ISBN_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Special:BookSources/978-1-4614-3809-0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Special:BookSources/978-1-4614-3809-0">978-1-4614-3809-0</a>)</li> <li>George E. Andrews and Bruce C. Berndt, Ramanujan's Lost Notebook: Part IV, (Springer, 2013, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="https://en-m-wikipedia-org.translate.goog/wiki/ISBN_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Special:BookSources/978-1-4614-4080-2?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Special:BookSources/978-1-4614-4080-2">978-1-4614-4080-2</a>)</li> <li>George E. Andrews and Bruce C. Berndt, Ramanujan's Lost Notebook: Part V, (Springer, 2018, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="https://en-m-wikipedia-org.translate.goog/wiki/ISBN_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Special:BookSources/978-3-319-77832-7?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Special:BookSources/978-3-319-77832-7">978-3-319-77832-7</a>)</li> <li>M. P. Chaudhary, A simple solution of some integrals given by Srinivasa Ramanujan, (Resonance: J. Sci. Education – publication of Indian Academy of Science, 2008)<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-166">[165]</a></li> <li>M.P. Chaudhary, Mock theta functions to mock theta conjectures, SCIENTIA, Series A: Math. Sci., (22)(2012) 33–46.</li> <li>M.P. Chaudhary, On modular relations for the Roger-Ramanujan type identities, Pacific J. Appl. Math., 7(3)(2016) 177–184.</li> </ul> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(14)"> <h2 id="Selected_publications_on_Ramanujan_and_his_work">Selected publications on Ramanujan and his work</h2> </div> <section class="mf-section-14 collapsible-block" id="mf-section-14"> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style> <div class="refbegin refbegin-columns references-column-width" style="column-width: 30em"> <ul> <li> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBerndt1998" class="citation book cs1">Berndt, Bruce C. (1998). Butzer, P. L.; Oberschelp, W.; Jongen, H. Th. (eds.). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://www.math.uiuc.edu/~berndt/articles/aachen.pdf">Charlemagne and His Heritage: 1200 Years of Civilization and Science in Europe</a> (PDF). Turnhout, Belgium: Brepols Verlag. pp. 119–146. <a href="https://en-m-wikipedia-org.translate.goog/wiki/ISBN_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Special:BookSources/978-2-503-50673-9?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Special:BookSources/978-2-503-50673-9"><bdi>978-2-503-50673-9</bdi></a>. <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://web.archive.org/web/20040909045617/http://www.math.uiuc.edu/~berndt/articles/aachen.pdf">Archived</a> (PDF) from the original on 9 September 2004.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Charlemagne+and+His+Heritage%3A+1200+Years+of+Civilization+and+Science+in+Europe&rft.place=Turnhout%2C+Belgium&rft.pages=119-146&rft.pub=Brepols+Verlag&rft.date=1998&rft.isbn=978-2-503-50673-9&rft.aulast=Berndt&rft.aufirst=Bruce+C.&rft_id=https%3A%2F%2Fwww.math.uiuc.edu%2F~berndt%2Farticles%2Faachen.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASrinivasa+Ramanujan" class="Z3988"></li> <li> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBerndtRankin1995" class="citation book cs1">Berndt, Bruce C.; Rankin, Robert A. (1995). Ramanujan: Letters and Commentary. Vol. 9. Providence, Rhode Island: <a href="https://en-m-wikipedia-org.translate.goog/wiki/American_Mathematical_Society?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="American Mathematical Society">American Mathematical Society</a>. <a href="https://en-m-wikipedia-org.translate.goog/wiki/ISBN_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Special:BookSources/978-0-8218-0287-8?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Special:BookSources/978-0-8218-0287-8"><bdi>978-0-8218-0287-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Ramanujan%3A+Letters+and+Commentary&rft.place=Providence%2C+Rhode+Island&rft.pub=American+Mathematical+Society&rft.date=1995&rft.isbn=978-0-8218-0287-8&rft.aulast=Berndt&rft.aufirst=Bruce+C.&rft.au=Rankin%2C+Robert+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASrinivasa+Ramanujan" class="Z3988"></li> <li> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBerndtRankin2001" class="citation book cs1"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Bruce_C._Berndt?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Bruce C. Berndt">Berndt, Bruce C.</a>; <a href="https://en-m-wikipedia-org.translate.goog/wiki/Robert_Alexander_Rankin?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Robert Alexander Rankin">Rankin, Robert A.</a> (2001). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://archive.org/details/trent_0116405145958">Ramanujan: Essays and Surveys</a>. Vol. 22. Providence, Rhode Island: <a href="https://en-m-wikipedia-org.translate.goog/wiki/American_Mathematical_Society?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="American Mathematical Society">American Mathematical Society</a>. <a href="https://en-m-wikipedia-org.translate.goog/wiki/ISBN_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Special:BookSources/978-0-8218-2624-9?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Special:BookSources/978-0-8218-2624-9"><bdi>978-0-8218-2624-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Ramanujan%3A+Essays+and+Surveys&rft.place=Providence%2C+Rhode+Island&rft.pub=American+Mathematical+Society&rft.date=2001&rft.isbn=978-0-8218-2624-9&rft.aulast=Berndt&rft.aufirst=Bruce+C.&rft.au=Rankin%2C+Robert+A.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Ftrent_0116405145958&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASrinivasa+Ramanujan" class="Z3988"></li> <li> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBerndt2006" class="citation book cs1">Berndt, Bruce C. 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Providence, Rhode Island: <a href="https://en-m-wikipedia-org.translate.goog/wiki/American_Mathematical_Society?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="American Mathematical Society">American Mathematical Society</a>. <a href="https://en-m-wikipedia-org.translate.goog/wiki/ISBN_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Special:BookSources/978-0-8218-4178-5?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Special:BookSources/978-0-8218-4178-5"><bdi>978-0-8218-4178-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Number+Theory+in+the+Spirit+of+Ramanujan&rft.place=Providence%2C+Rhode+Island&rft.pub=American+Mathematical+Society&rft.date=2006&rft.isbn=978-0-8218-4178-5&rft.aulast=Berndt&rft.aufirst=Bruce+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASrinivasa+Ramanujan" class="Z3988"></li> <li> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBerndt1985" class="citation book cs1">Berndt, Bruce C. (1985). Ramanujan's Notebooks: Part I. New York: Springer. <a href="https://en-m-wikipedia-org.translate.goog/wiki/ISBN_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Special:BookSources/978-0-387-96110-1?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Special:BookSources/978-0-387-96110-1"><bdi>978-0-387-96110-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Ramanujan%27s+Notebooks%3A+Part+I&rft.place=New+York&rft.pub=Springer&rft.date=1985&rft.isbn=978-0-387-96110-1&rft.aulast=Berndt&rft.aufirst=Bruce+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASrinivasa+Ramanujan" class="Z3988"></li> <li> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBerndt1999" class="citation book cs1">Berndt, Bruce C. (1999). Ramanujan's Notebooks: Part II. New York: Springer. <a href="https://en-m-wikipedia-org.translate.goog/wiki/ISBN_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Special:BookSources/978-0-387-96794-3?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Special:BookSources/978-0-387-96794-3"><bdi>978-0-387-96794-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Ramanujan%27s+Notebooks%3A+Part+II&rft.place=New+York&rft.pub=Springer&rft.date=1999&rft.isbn=978-0-387-96794-3&rft.aulast=Berndt&rft.aufirst=Bruce+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASrinivasa+Ramanujan" class="Z3988"></li> <li> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBerndt2004" class="citation book cs1">Berndt, Bruce C. (2004). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://archive.org/details/ramanujansnotebo0000rama">Ramanujan's Notebooks: Part III</a>. New York: Springer. <a href="https://en-m-wikipedia-org.translate.goog/wiki/ISBN_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Special:BookSources/978-0-387-97503-0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Special:BookSources/978-0-387-97503-0"><bdi>978-0-387-97503-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Ramanujan%27s+Notebooks%3A+Part+III&rft.place=New+York&rft.pub=Springer&rft.date=2004&rft.isbn=978-0-387-97503-0&rft.aulast=Berndt&rft.aufirst=Bruce+C.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Framanujansnotebo0000rama&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASrinivasa+Ramanujan" class="Z3988"></li> <li> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBerndt1993" class="citation book cs1">Berndt, Bruce C. (1993). Ramanujan's Notebooks: Part IV. New York: Springer. <a href="https://en-m-wikipedia-org.translate.goog/wiki/ISBN_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Special:BookSources/978-0-387-94109-7?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Special:BookSources/978-0-387-94109-7"><bdi>978-0-387-94109-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Ramanujan%27s+Notebooks%3A+Part+IV&rft.place=New+York&rft.pub=Springer&rft.date=1993&rft.isbn=978-0-387-94109-7&rft.aulast=Berndt&rft.aufirst=Bruce+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASrinivasa+Ramanujan" class="Z3988"></li> <li> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBerndt2005" class="citation book cs1">Berndt, Bruce C. (2005). Ramanujan's Notebooks: Part V. New York: Springer. <a href="https://en-m-wikipedia-org.translate.goog/wiki/ISBN_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Special:BookSources/978-0-387-94941-3?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Special:BookSources/978-0-387-94941-3"><bdi>978-0-387-94941-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Ramanujan%27s+Notebooks%3A+Part+V&rft.place=New+York&rft.pub=Springer&rft.date=2005&rft.isbn=978-0-387-94941-3&rft.aulast=Berndt&rft.aufirst=Bruce+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASrinivasa+Ramanujan" class="Z3988"></li> <li> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHardy1937" class="citation journal cs1">Hardy, G. H. (March 1937). "The Indian Mathematician Ramanujan". The American Mathematical Monthly. 44 (3): 137–155. <a href="https://en-m-wikipedia-org.translate.goog/wiki/Doi_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://doi.org/10.2307%252F2301659">10.2307/2301659</a>. <a href="https://en-m-wikipedia-org.translate.goog/wiki/JSTOR_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://www.jstor.org/stable/2301659">2301659</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+American+Mathematical+Monthly&rft.atitle=The+Indian+Mathematician+Ramanujan&rft.volume=44&rft.issue=3&rft.pages=137-155&rft.date=1937-03&rft_id=info%3Adoi%2F10.2307%2F2301659&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2301659%23id-name%3DJSTOR&rft.aulast=Hardy&rft.aufirst=G.+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASrinivasa+Ramanujan" class="Z3988"></li> <li> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHardy1978" class="citation book cs1">Hardy, G. H. (1978). Ramanujan. New York: Chelsea Pub. Co. <a href="https://en-m-wikipedia-org.translate.goog/wiki/ISBN_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Special:BookSources/978-0-8284-0136-4?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Special:BookSources/978-0-8284-0136-4"><bdi>978-0-8284-0136-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Ramanujan&rft.place=New+York&rft.pub=Chelsea+Pub.+Co.&rft.date=1978&rft.isbn=978-0-8284-0136-4&rft.aulast=Hardy&rft.aufirst=G.+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASrinivasa+Ramanujan" class="Z3988"></li> <li> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHardy1999" class="citation book cs1">Hardy, G. H. (1999). Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work. Providence, Rhode Island: American Mathematical Society. <a href="https://en-m-wikipedia-org.translate.goog/wiki/ISBN_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Special:BookSources/978-0-8218-2023-0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Special:BookSources/978-0-8218-2023-0"><bdi>978-0-8218-2023-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Ramanujan%3A+Twelve+Lectures+on+Subjects+Suggested+by+His+Life+and+Work&rft.place=Providence%2C+Rhode+Island&rft.pub=American+Mathematical+Society&rft.date=1999&rft.isbn=978-0-8218-2023-0&rft.aulast=Hardy&rft.aufirst=G.+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASrinivasa+Ramanujan" class="Z3988"></li> <li> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHenderson1995" class="citation book cs1">Henderson, Harry (1995). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://archive.org/details/modernmathematic00hend">Modern Mathematicians</a>. New York: Facts on File Inc. <a href="https://en-m-wikipedia-org.translate.goog/wiki/ISBN_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Special:BookSources/978-0-8160-3235-8?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Special:BookSources/978-0-8160-3235-8"><bdi>978-0-8160-3235-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Modern+Mathematicians&rft.place=New+York&rft.pub=Facts+on+File+Inc.&rft.date=1995&rft.isbn=978-0-8160-3235-8&rft.aulast=Henderson&rft.aufirst=Harry&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fmodernmathematic00hend&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASrinivasa+Ramanujan" class="Z3988"></li> <li> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKanigel1991" class="citation book cs1"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Robert_Kanigel?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Robert Kanigel">Kanigel, Robert</a> (1991). <a href="https://en-m-wikipedia-org.translate.goog/wiki/The_Man_Who_Knew_Infinity:_a_Life_of_the_Genius_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="The Man Who Knew Infinity: a Life of the Genius Ramanujan">The Man Who Knew Infinity: a Life of the Genius Ramanujan</a>. 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London: Bloomsbury. <a href="https://en-m-wikipedia-org.translate.goog/wiki/ISBN_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Special:BookSources/978-0-7475-9370-6?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Special:BookSources/978-0-7475-9370-6"><bdi>978-0-7475-9370-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Indian+Clerk&rft.place=London&rft.edition=paperback&rft.pub=Bloomsbury&rft.date=2007&rft.isbn=978-0-7475-9370-6&rft.aulast=Leavitt&rft.aufirst=David&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASrinivasa+Ramanujan" class="Z3988"></li> <li> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNarlikar2003" class="citation book cs1"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Jayant_V._Narlikar?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Jayant V. Narlikar">Narlikar, Jayant V.</a> (2003). Scientific Edge: the Indian Scientist From Vedic to Modern Times. New Delhi, India: Penguin Books. <a href="https://en-m-wikipedia-org.translate.goog/wiki/ISBN_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Special:BookSources/978-0-14-303028-7?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Special:BookSources/978-0-14-303028-7"><bdi>978-0-14-303028-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Scientific+Edge%3A+the+Indian+Scientist+From+Vedic+to+Modern+Times&rft.place=New+Delhi%2C+India&rft.pub=Penguin+Books&rft.date=2003&rft.isbn=978-0-14-303028-7&rft.aulast=Narlikar&rft.aufirst=Jayant+V.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASrinivasa+Ramanujan" class="Z3988"></li> <li> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFOnoAczel2016" class="citation book cs1"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Ken_Ono?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Ken Ono">Ono, Ken</a>; <a href="https://en-m-wikipedia-org.translate.goog/wiki/Amir_Aczel?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Amir Aczel">Aczel, Amir D.</a> (13 April 2016). My Search for Ramanujan: How I Learned to Count. <a href="https://en-m-wikipedia-org.translate.goog/wiki/Springer_Science%2BBusiness_Media?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Springer Science+Business Media">Springer</a>. <a href="https://en-m-wikipedia-org.translate.goog/wiki/ISBN_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Special:BookSources/978-3319255668?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Special:BookSources/978-3319255668"><bdi>978-3319255668</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=My+Search+for+Ramanujan%3A+How+I+Learned+to+Count&rft.pub=Springer&rft.date=2016-04-13&rft.isbn=978-3319255668&rft.aulast=Ono&rft.aufirst=Ken&rft.au=Aczel%2C+Amir+D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASrinivasa+Ramanujan" class="Z3988"></li> <li> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSankaran2005" class="citation report cs1 cs1-prop-foreign-lang-source">Sankaran, T. M. (2005). Srinivasa Ramanujan- Ganitha lokathile Mahaprathibha (Report) (in Malayalam). Kochi, India: <a href="https://en-m-wikipedia-org.translate.goog/wiki/Kerala_Sasthra_Sahithya_Parishad?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Kerala Sasthra Sahithya Parishad">Kerala Sasthra Sahithya Parishad</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=report&rft.btitle=Srinivasa+Ramanujan-+Ganitha+lokathile+Mahaprathibha&rft.place=Kochi%2C+India&rft.pub=Kerala+Sasthra+Sahithya+Parishad&rft.date=2005&rft.aulast=Sankaran&rft.aufirst=T.+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASrinivasa+Ramanujan" class="Z3988"></li> </ul> </div> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(15)"> <h2 id="Selected_publications_on_works_of_Ramanujan">Selected publications on works of Ramanujan</h2> </div> <section class="mf-section-15 collapsible-block" id="mf-section-15"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239549316"> <div class="refbegin refbegin-columns references-column-width" style="column-width: 30em"> <ul> <li> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRamanujanHardySeshu_AiyarWilson2000" class="citation book cs1">Ramanujan, Srinivasa; Hardy, G. H.; Seshu Aiyar, P. V.; <a href="https://en-m-wikipedia-org.translate.goog/wiki/Bertram_Martin_Wilson?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Bertram Martin Wilson">Wilson, B. M.</a>; Berndt, Bruce C. (2000). Collected Papers of Srinivasa Ramanujan. AMS. <a href="https://en-m-wikipedia-org.translate.goog/wiki/ISBN_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Special:BookSources/978-0-8218-2076-6?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Special:BookSources/978-0-8218-2076-6"><bdi>978-0-8218-2076-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Collected+Papers+of+Srinivasa+Ramanujan&rft.pub=AMS&rft.date=2000&rft.isbn=978-0-8218-2076-6&rft.aulast=Ramanujan&rft.aufirst=Srinivasa&rft.au=Hardy%2C+G.+H.&rft.au=Seshu+Aiyar%2C+P.+V.&rft.au=Wilson%2C+B.+M.&rft.au=Berndt%2C+Bruce+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASrinivasa+Ramanujan" class="Z3988"></li> </ul> <dl> <dd> This book was originally published in 1927<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-167">[166]</a> 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. </dd> </dl> <ul> <li> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFS._Ramanujan1957" class="citation book cs1">S. Ramanujan (1957). Notebooks (2 Volumes). Bombay: Tata Institute of Fundamental Research.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Notebooks+%282+Volumes%29&rft.place=Bombay&rft.pub=Tata+Institute+of+Fundamental+Research&rft.date=1957&rft.au=S.+Ramanujan&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASrinivasa+Ramanujan" class="Z3988"></li> </ul> <dl> <dd> These books contain photocopies of the original notebooks as written by Ramanujan. </dd> </dl> <ul> <li> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFS._Ramanujan1988" class="citation book cs1">S. Ramanujan (1988). The Lost Notebook and Other Unpublished Papers. New Delhi: Narosa. <a href="https://en-m-wikipedia-org.translate.goog/wiki/ISBN_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Special:BookSources/978-3-540-18726-4?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Special:BookSources/978-3-540-18726-4"><bdi>978-3-540-18726-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Lost+Notebook+and+Other+Unpublished+Papers&rft.place=New+Delhi&rft.pub=Narosa&rft.date=1988&rft.isbn=978-3-540-18726-4&rft.au=S.+Ramanujan&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASrinivasa+Ramanujan" class="Z3988"></li> </ul> <dl> <dd> This book contains photocopies of the pages of the "Lost Notebook". </dd> </dl> <ul> <li><a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://www.imsc.res.in/~rao/ramanujan/collectedpapers/question/qJIMS.htm">Problems posed by Ramanujan</a>, Journal of the Indian Mathematical Society.</li> <li> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFS._Ramanujan2012" class="citation book cs1">S. Ramanujan (2012). Notebooks (2 Volumes). Bombay: Tata Institute of Fundamental Research.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Notebooks+%282+Volumes%29&rft.place=Bombay&rft.pub=Tata+Institute+of+Fundamental+Research&rft.date=2012&rft.au=S.+Ramanujan&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASrinivasa+Ramanujan" class="Z3988"></li> </ul> <dl> <dd> This was produced from scanned and microfilmed images of the original manuscripts by expert archivists of Roja Muthiah Research Library, Chennai. </dd> </dl> </div> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(16)"> <h2 id="See_also">See also</h2> </div> <section class="mf-section-16 collapsible-block" id="mf-section-16"> <style data-mw-deduplicate="TemplateStyles:r1259569809">.mw-parser-output .portalbox{padding:0;margin:0.5em 0;display:table;box-sizing:border-box;max-width:175px;list-style:none}.mw-parser-output .portalborder{border:1px solid var(--border-color-base,#a2a9b1);padding:0.1em;background:var(--background-color-neutral-subtle,#f8f9fa)}.mw-parser-output .portalbox-entry{display:table-row;font-size:85%;line-height:110%;height:1.9em;font-style:italic;font-weight:bold}.mw-parser-output .portalbox-image{display:table-cell;padding:0.2em;vertical-align:middle;text-align:center}.mw-parser-output .portalbox-link{display:table-cell;padding:0.2em 0.2em 0.2em 0.3em;vertical-align:middle}@media(min-width:720px){.mw-parser-output .portalleft{clear:left;float:left;margin:0.5em 1em 0.5em 0}.mw-parser-output .portalright{clear:right;float:right;margin:0.5em 0 0.5em 1em}}</style> <ul role="navigation" aria-label="Portals" class="noprint portalbox portalborder portalright"> <li class="portalbox-entry"><a href="https://en-m-wikipedia-org.translate.goog/wiki/File:Nuvola_apps_edu_mathematics_blue-p.svg?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-file-description"> <noscript> <img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/28px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="28" height="28" class="mw-file-element" data-file-width="128" data-file-height="128"> </noscript><span class="lazy-image-placeholder" style="width: 28px;height: 28px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/28px-Nuvola_apps_edu_mathematics_blue-p.svg.png" data-alt="icon" data-width="28" data-height="28" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/42px-Nuvola_apps_edu_mathematics_blue-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/56px-Nuvola_apps_edu_mathematics_blue-p.svg.png 2x" data-class="mw-file-element"> </a><a href="https://en-m-wikipedia-org.translate.goog/wiki/Portal:Mathematics?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Portal:Mathematics">Mathematics portal</a></li> <li class="portalbox-entry"> <noscript> <img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/6/69/P_vip.svg/28px-P_vip.svg.png" decoding="async" width="28" height="28" class="mw-file-element" data-file-width="1911" data-file-height="1944"> </noscript><span class="lazy-image-placeholder" style="width: 28px;height: 28px;" data-src="//upload.wikimedia.org/wikipedia/en/thumb/6/69/P_vip.svg/28px-P_vip.svg.png" data-alt="" data-width="28" data-height="28" data-srcset="//upload.wikimedia.org/wikipedia/en/thumb/6/69/P_vip.svg/41px-P_vip.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/6/69/P_vip.svg/55px-P_vip.svg.png 2x" data-class="mw-file-element"> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Portal:Biography?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Portal:Biography">Biography portal</a></li> <li class="portalbox-entry"> <noscript> <img alt="flag" src="//upload.wikimedia.org/wikipedia/en/thumb/4/41/Flag_of_India.svg/32px-Flag_of_India.svg.png" decoding="async" width="32" height="21" class="mw-file-element" data-file-width="900" data-file-height="600"> </noscript><span class="lazy-image-placeholder" style="width: 32px;height: 21px;" data-src="//upload.wikimedia.org/wikipedia/en/thumb/4/41/Flag_of_India.svg/32px-Flag_of_India.svg.png" data-alt="flag" data-width="32" data-height="21" data-srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/41/Flag_of_India.svg/48px-Flag_of_India.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/41/Flag_of_India.svg/64px-Flag_of_India.svg.png 2x" data-class="mw-file-element"> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Portal:India?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Portal:India">India portal</a></li> </ul> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1184024115"> <div class="div-col"> <ul> <li><a href="https://en-m-wikipedia-org.translate.goog/wiki/1729_(number)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="1729 (number)">1729 (number)</a></li> <li><a href="https://en-m-wikipedia-org.translate.goog/wiki/Brown_numbers?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Brown numbers">Brown numbers</a></li> <li><a href="https://en-m-wikipedia-org.translate.goog/wiki/List_of_amateur_mathematicians?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="List of amateur mathematicians">List of amateur mathematicians</a></li> <li><a href="https://en-m-wikipedia-org.translate.goog/wiki/List_of_Indian_mathematicians?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="List of Indian mathematicians">List of Indian mathematicians</a></li> <li><a href="https://en-m-wikipedia-org.translate.goog/wiki/Ramanujan_graph?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Ramanujan graph">Ramanujan graph</a></li> <li><a href="https://en-m-wikipedia-org.translate.goog/wiki/Ramanujan_summation?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Ramanujan summation">Ramanujan summation</a></li> <li><a href="https://en-m-wikipedia-org.translate.goog/wiki/Ramanujan%27s_constant?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Ramanujan's constant">Ramanujan's constant</a></li> <li><a href="https://en-m-wikipedia-org.translate.goog/wiki/Ramanujan%27s_ternary_quadratic_form?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Ramanujan's ternary quadratic form">Ramanujan's ternary quadratic form</a></li> <li><a href="https://en-m-wikipedia-org.translate.goog/wiki/Rank_of_a_partition?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Rank of a partition">Rank of a partition</a></li> </ul> </div> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(17)"> <h2 id="Footnotes">Footnotes</h2> </div> <section class="mf-section-17 collapsible-block" id="mf-section-17"> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style> <div class="reflist reflist-lower-alpha"> <div class="mw-references-wrap"> <ol class="references"> <li id="cite_note-4"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-4">^</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Fellow_of_the_Royal_Society?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Fellow of the Royal Society">FRS</a> (<a href="https://en-m-wikipedia-org.translate.goog/wiki/Help:IPA/English?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Help:IPA/English">/ˈsriːnɪvɑːsə rɑːˈmɑːnʊdʒən/</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Help:Pronunciation_respelling_key?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Help:Pronunciation respelling key">SREE-nih-vah-sə rah-MAH-nuuj-ən</a>;<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-1">[1]</a> born Srinivasa Ramanujan Aiyangar, <style data-mw-deduplicate="TemplateStyles:r1177148991">.mw-parser-output .IPA-label-small{font-size:85%}.mw-parser-output .references .IPA-label-small,.mw-parser-output .infobox .IPA-label-small,.mw-parser-output .navbox .IPA-label-small{font-size:100%}</style>Tamil: <a href="https://en-m-wikipedia-org.translate.goog/wiki/Help:IPA/Tamil?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Help:IPA/Tamil">[sriːniʋaːsa ɾaːmaːnud͡ʑan ajːaŋgar]</a>)<a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-ODNB-Ramanujan-2">[2]</a><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-3">[3]</a></li> </ol> </div> </div> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(18)"> <h2 id="References">References</h2> </div> <section class="mf-section-18 collapsible-block" id="mf-section-18"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"> <div class="reflist"> <div class="mw-references-wrap mw-references-columns"> <ol class="references"> <li id="cite_note-1"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-1">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFOlaussonSangster2006" class="citation book cs1">Olausson, Lena; Sangster, Catherine (2006). Oxford BBC Guide to Pronunciation. Oxford University Press. p. 322. <a href="https://en-m-wikipedia-org.translate.goog/wiki/ISBN_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Special:BookSources/978-0-19-280710-6?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Special:BookSources/978-0-19-280710-6"><bdi>978-0-19-280710-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Oxford+BBC+Guide+to+Pronunciation&rft.pages=322&rft.pub=Oxford+University+Press&rft.date=2006&rft.isbn=978-0-19-280710-6&rft.aulast=Olausson&rft.aufirst=Lena&rft.au=Sangster%2C+Catherine&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASrinivasa+Ramanujan" class="Z3988"></li> <li id="cite_note-ODNB-Ramanujan-2">^ <a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-ODNB-Ramanujan_2-0">a</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-ODNB-Ramanujan_2-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKanigel2004" class="citation encyclopaedia cs1">Kanigel, Robert (2004). "Ramanujan, Srinivasa". <a href="https://en-m-wikipedia-org.translate.goog/wiki/Dictionary_of_National_Biography?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Oxford_Dictionary_of_National_Biography" title="Dictionary of National Biography">Oxford Dictionary of National Biography</a> (online ed.). Oxford University Press. <a href="https://en-m-wikipedia-org.translate.goog/wiki/Doi_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://doi.org/10.1093%252Fref%253Aodnb%252F51582">10.1093/ref:odnb/51582</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Ramanujan%2C+Srinivasa&rft.btitle=Oxford+Dictionary+of+National+Biography&rft.edition=online&rft.pub=Oxford+University+Press&rft.date=2004&rft_id=info%3Adoi%2F10.1093%2Fref%3Aodnb%2F51582&rft.aulast=Kanigel&rft.aufirst=Robert&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASrinivasa+Ramanujan" class="Z3988"> (Subscription or <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://www.oxforddnb.com/help/subscribe%23public">UK public library membership</a> required.)</li> <li id="cite_note-3"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-3">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://trove.nla.gov.au/people/895585?c%3Dpeople">"Ramanujan Aiyangar, Srinivasa (1887–1920)"</a>. trove.nla.gov.au.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=trove.nla.gov.au&rft.atitle=Ramanujan+Aiyangar%2C+Srinivasa+%281887%E2%80%931920%29&rft_id=https%3A%2F%2Ftrove.nla.gov.au%2Fpeople%2F895585%3Fc%3Dpeople&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASrinivasa+Ramanujan" class="Z3988"></li> <li id="cite_note-5"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-5">^</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Hans_Eysenck?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Hans Eysenck">Hans Eysenck</a> (1995). Genius, p. 197. Cambridge University Press, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="https://en-m-wikipedia-org.translate.goog/wiki/ISBN_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Special:BookSources/0-521-48508-8?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Special:BookSources/0-521-48508-8">0-521-48508-8</a>.</li> <li id="cite_note-6"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-6">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHardy1940" class="citation book cs1">Hardy, Godfrey Harold (1940). Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work. Cambridge University Press. p. 9. <a href="https://en-m-wikipedia-org.translate.goog/wiki/ISBN_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Special:BookSources/0-8218-2023-0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Special:BookSources/0-8218-2023-0"><bdi>0-8218-2023-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Ramanujan%3A+Twelve+Lectures+on+Subjects+Suggested+by+His+Life+and+Work&rft.pages=9&rft.pub=Cambridge+University+Press&rft.date=1940&rft.isbn=0-8218-2023-0&rft.aulast=Hardy&rft.aufirst=Godfrey+Harold&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASrinivasa+Ramanujan" class="Z3988"></li> <li id="cite_note-7"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-7">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBerndt1997" class="citation book cs1 cs1-prop-long-vol">Berndt, Bruce C. (12 December 1997). Ramanujan's Notebooks. Vol. Part 5. Springer Science & Business. p. 4. <a href="https://en-m-wikipedia-org.translate.goog/wiki/ISBN_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Special:BookSources/978-0-38794941-3?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Special:BookSources/978-0-38794941-3"><bdi>978-0-38794941-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Ramanujan%27s+Notebooks&rft.pages=4&rft.pub=Springer+Science+%26+Business&rft.date=1997-12-12&rft.isbn=978-0-38794941-3&rft.aulast=Berndt&rft.aufirst=Bruce+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASrinivasa+Ramanujan" class="Z3988"></li> <li id="cite_note-ono-8">^ <a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-ono_8-0">a</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-ono_8-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFOno2006" class="citation journal cs1"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Ken_Ono?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Ken Ono">Ono, Ken</a> (June–July 2006). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://www.ams.org/notices/200606/fea-ono.pdf">"Honoring a Gift from Kumbakonam"</a> (PDF). <a href="https://en-m-wikipedia-org.translate.goog/wiki/Notices_of_the_American_Mathematical_Society?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Notices of the American Mathematical Society">Notices of the American Mathematical Society</a>. 53 (6): 640–51 [649–50]. <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://web.archive.org/web/20070621074858/https://www.ams.org/notices/200606/fea-ono.pdf">Archived</a> (PDF) from the original on 21 June 2007. Retrieved 23 June 2007.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Notices+of+the+American+Mathematical+Society&rft.atitle=Honoring+a+Gift+from+Kumbakonam&rft.volume=53&rft.issue=6&rft.pages=640-51+649-50&rft.date=2006-06%2F2006-07&rft.aulast=Ono&rft.aufirst=Ken&rft_id=https%3A%2F%2Fwww.ams.org%2Fnotices%2F200606%2Ffea-ono.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASrinivasa+Ramanujan" class="Z3988"></li> <li id="cite_note-frontline-9">^ <a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-frontline_9-0">a</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-frontline_9-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation journal cs1"><a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://web.archive.org/web/20130925201456/http://www.frontline.in/static/html/fl1617/16170810.htm">"Rediscovering Ramanujan"</a>. <a href="https://en-m-wikipedia-org.translate.goog/wiki/Frontline_(magazine)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Frontline (magazine)">Frontline</a>. 16 (17): 650. August 1999. Archived from <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://www.frontline.in/static/html/fl1617/16170810.htm">the original</a> on 25 September 2013. Retrieved 20 December 2012.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Frontline&rft.atitle=Rediscovering+Ramanujan&rft.volume=16&rft.issue=17&rft.pages=650&rft.date=1999-08&rft_id=https%3A%2F%2Fwww.frontline.in%2Fstatic%2Fhtml%2Ffl1617%2F16170810.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASrinivasa+Ramanujan" class="Z3988"></li> <li id="cite_note-10"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-10">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAlladiElliottGranville1998" class="citation book cs1">Alladi, Krishnaswami; Elliott, P. D. T. A.; Granville, A. (30 September 1998). Analytic and Elementary Number Theory: A Tribute to Mathematical Legend Paul Erdos. Springer Science & Business. p. 6. <a href="https://en-m-wikipedia-org.translate.goog/wiki/ISBN_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Special:BookSources/978-0-79238273-7?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Special:BookSources/978-0-79238273-7"><bdi>978-0-79238273-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Analytic+and+Elementary+Number+Theory%3A+A+Tribute+to+Mathematical+Legend+Paul+Erdos&rft.pages=6&rft.pub=Springer+Science+%26+Business&rft.date=1998-09-30&rft.isbn=978-0-79238273-7&rft.aulast=Alladi&rft.aufirst=Krishnaswami&rft.au=Elliott%2C+P.+D.+T.+A.&rft.au=Granville%2C+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASrinivasa+Ramanujan" class="Z3988"></li> <li id="cite_note-ono_simpleproperties-11"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-ono_simpleproperties_11-0">^</a> <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://www.newscientist.com/article/dn20039-deep-meaning-in-ramanujans-simple-pattern/">Deep meaning in Ramanujan's 'simple' pattern</a> <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://web.archive.org/web/20170803140127/https://www.newscientist.com/article/dn20039-deep-meaning-in-ramanujans-simple-pattern/">Archived</a> 3 August 2017 at the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Wayback_Machine?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Wayback Machine">Wayback Machine</a></li> <li id="cite_note-ono_mock-12"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-ono_mock_12-0">^</a> <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://www.newscientist.com/article/mg21628904-200-mathematical-proof-reveals-magic-of-ramanujans-genius/">"Mathematical proof reveals magic of Ramanujan's genius"</a> <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://web.archive.org/web/20170709084351/https://www.newscientist.com/article/mg21628904-200-mathematical-proof-reveals-magic-of-ramanujans-genius/">Archived</a> 9 July 2017 at the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Wayback_Machine?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Wayback Machine">Wayback Machine</a>. <a href="https://en-m-wikipedia-org.translate.goog/wiki/New_Scientist?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="New Scientist">New Scientist</a>.</li> <li id="cite_note-Kanigel-1991-12-13">^ <a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Kanigel-1991-12_13-0">a</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Kanigel-1991-12_13-1">b</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#CITEREFKanigel1991">Kanigel 1991</a>, p. 12</li> <li id="cite_note-14"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-14">^</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#CITEREFKanigel1991">Kanigel 1991</a>, p. 11</li> <li id="cite_note-15"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-15">^</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#CITEREFKanigel1991">Kanigel 1991</a>, pp. 17–18</li> <li id="cite_note-16"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-16">^</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#CITEREFBerndtRankin2001">Berndt & Rankin 2001</a>, p. 89</li> <li id="cite_note-hindu-17"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-hindu_17-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSrinivasan2012" class="citation news cs1">Srinivasan, Pankaja (19 October 2012). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://www.thehindu.com/features/friday-review/history-and-culture/the-nostalgia-formula/article4013282.ece">"The Nostalgia Formula"</a>. The Hindu. 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Maths History. Retrieved 29 October 2022.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Maths+History&rft.atitle=Srinivasa+Ramanujan+-+Biography&rft_id=https%3A%2F%2Fmathshistory.st-andrews.ac.uk%2FBiographies%2FRamanujan%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASrinivasa+Ramanujan" class="Z3988"></li> <li id="cite_note-27"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-27">^</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#CITEREFKanigel1991">Kanigel 1991</a>, p. 39</li> <li id="cite_note-28"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-28">^</a> McElroy, Tucker (2005). A to Z of mathematicians. 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Veṅkatesvara Seshu (2000), "Collected papers of Srinivasa Ramanujan", Nature, 123 (3104): xii, <a href="https://en-m-wikipedia-org.translate.goog/wiki/Bibcode_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ui.adsabs.harvard.edu/abs/1929Natur.123..631L">1929Natur.123..631L</a>, <a href="https://en-m-wikipedia-org.translate.goog/wiki/Doi_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://doi.org/10.1038%252F123631a0">10.1038/123631a0</a>, <a href="https://en-m-wikipedia-org.translate.goog/wiki/ISBN_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Special:BookSources/978-0-8218-2076-6?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Special:BookSources/978-0-8218-2076-6"><bdi>978-0-8218-2076-6</bdi></a>, <a href="https://en-m-wikipedia-org.translate.goog/wiki/S2CID_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://api.semanticscholar.org/CorpusID:44812911">44812911</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nature&rft.atitle=Collected+papers+of+Srinivasa+Ramanujan&rft.volume=123&rft.issue=3104&rft.pages=xii&rft.date=2000&rft_id=info%3Adoi%2F10.1038%2F123631a0&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A44812911%23id-name%3DS2CID&rft_id=info%3Abibcode%2F1929Natur.123..631L&rft.isbn=978-0-8218-2076-6&rft.aulast=Ramanujan+Aiyangar&rft.aufirst=Srinivasa&rft.au=Hardy%2C+Godfrey+Harold&rft.au=Aiyar%2C+P.+Ve%E1%B9%85katesvara+Seshu&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASrinivasa+Ramanujan" class="Z3988"></li> <li id="cite_note-30"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-30">^</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#CITEREFKanigel1991">Kanigel 1991</a>, p. 90</li> <li id="cite_note-Kanigel-1991-unknown-page-31">^ <a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Kanigel-1991-unknown-page_31-0">a</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Kanigel-1991-unknown-page_31-1">b</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#CITEREFKanigel1991">Kanigel 1991</a>, p. ??</li> <li id="cite_note-Kanigel-1991-28-32"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Kanigel-1991-28_32-0">^</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#CITEREFKanigel1991">Kanigel 1991</a>, p. 28</li> <li id="cite_note-Kanigel-1991-45-33"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Kanigel-1991-45_33-0">^</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#CITEREFKanigel1991">Kanigel 1991</a>, p. 45</li> <li id="cite_note-Kanigel-1991-47-48-34">^ <a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Kanigel-1991-47-48_34-0">a</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Kanigel-1991-47-48_34-1">b</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#CITEREFKanigel1991">Kanigel 1991</a>, pp. 47–48</li> <li id="cite_note-35"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-35">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation news cs1"><a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://www.thehindu.com/features/friday-review/history-and-culture/ramanujan-lost-and-found-a-1905-letter-from-the-hindu/article2745164.ece">"Ramanujan lost and found: a 1905 letter from The Hindu"</a>. 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The Hindu. <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://web.archive.org/web/20170826080524/https://www.thehindu.com/features/friday-review/travails-of-a-genius/article4857108.ece">Archived</a> from the original on 26 August 2017. Retrieved 7 September 2016.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Hindu&rft.atitle=Travails+of+a+Genius&rft.date=2013-06-27&rft.aulast=Krishnamachari&rft.aufirst=Suganthi&rft_id=https%3A%2F%2Fwww.thehindu.com%2Ffeatures%2Ffriday-review%2Ftravails-of-a-genius%2Farticle4857108.ece&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASrinivasa+Ramanujan" class="Z3988"></li> <li id="cite_note-Kanigel-1991-55-56-37"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Kanigel-1991-55-56_37-0">^</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#CITEREFKanigel1991">Kanigel 1991</a>, p. 55–56</li> <li id="cite_note-krish-38"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-krish_38-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKrishnamurthy" class="citation web cs1">Krishnamurthy, V. <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://web.archive.org/web/20160921182431/https://www.krishnamurthys.com/profvk/VK2/Srinivasa_Ramanujan.html">"Srinivasa Ramanujan – His life and his genius"</a>. www.krishnamurthys.com. (Expository address delivered on Sep.16, 1987 at Visvesvarayya Auditorium as part of the celebrations of Ramanujan Centenary by the IISC, Bangalore). Archived from <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://www.krishnamurthys.com/profvk/VK2/Srinivasa_Ramanujan.html">the original</a> on 21 September 2016. Retrieved 7 September 2016.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.krishnamurthys.com&rft.atitle=Srinivasa+Ramanujan+%E2%80%93+His+life+and+his+genius&rft.aulast=Krishnamurthy&rft.aufirst=V.&rft_id=https%3A%2F%2Fwww.krishnamurthys.com%2Fprofvk%2FVK2%2FSrinivasa_Ramanujan.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASrinivasa+Ramanujan" class="Z3988"></li> <li id="cite_note-39"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-39">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://www.livemint.com/Leisure/6vJOM00P1sGZgCxV7RIasI/The-seamstress-and-the-mathematician.html">"The seamstress and the mathematician"</a>. Live mint. 20 April 2018.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Live+mint&rft.atitle=The+seamstress+and+the+mathematician&rft.date=2018-04-20&rft_id=https%3A%2F%2Fwww.livemint.com%2FLeisure%2F6vJOM00P1sGZgCxV7RIasI%2FThe-seamstress-and-the-mathematician.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASrinivasa+Ramanujan" class="Z3988"></li> <li id="cite_note-Kanigel-1991-71-40"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Kanigel-1991-71_40-0">^</a> ><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#CITEREFKanigel1991">Kanigel 1991</a>, p. 71</li> <li id="cite_note-History_in_adult_human_sexual_behavior_with_children_and_adolescents_in_Western_societies-41"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-History_in_adult_human_sexual_behavior_with_children_and_adolescents_in_Western_societies_41-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBullough1990" class="citation book cs1">Bullough, V.L. (1990). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://books.google.com/books?id%3DnnKEBwAAQBAJ%26pg%3DPA69">"2. History in adult human sexual behavior with children and adolescents in Western societies"</a>. Pedophilia: Biosocial Dimensions. New York: Springer-Verlag. p. 71. <a href="https://en-m-wikipedia-org.translate.goog/wiki/ISBN_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Special:BookSources/978-1-46139684-0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Special:BookSources/978-1-46139684-0"><bdi>978-1-46139684-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=2.+History+in+adult+human+sexual+behavior+with+children+and+adolescents+in+Western+societies&rft.btitle=Pedophilia%3A+Biosocial+Dimensions&rft.place=New+York&rft.pages=71&rft.pub=Springer-Verlag&rft.date=1990&rft.isbn=978-1-46139684-0&rft.aulast=Bullough&rft.aufirst=V.L.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DnnKEBwAAQBAJ%26pg%3DPA69&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASrinivasa+Ramanujan" class="Z3988"></li> <li id="cite_note-42"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-42">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKolata1987" class="citation journal cs1"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Gina_Kolata?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Gina Kolata">Kolata, Gina</a> (19 June 1987). "Remembering a 'Magical Genius'". Science. New Series. 236 (4808): 1519–21. <a href="https://en-m-wikipedia-org.translate.goog/wiki/Bibcode_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ui.adsabs.harvard.edu/abs/1987Sci...236.1519K">1987Sci...236.1519K</a>. <a href="https://en-m-wikipedia-org.translate.goog/wiki/Doi_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://doi.org/10.1126%252Fscience.236.4808.1519">10.1126/science.236.4808.1519</a>. <a href="https://en-m-wikipedia-org.translate.goog/wiki/PMID_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://pubmed.ncbi.nlm.nih.gov/17835731">17835731</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Science&rft.atitle=Remembering+a+%27Magical+Genius%27&rft.volume=236&rft.issue=4808&rft.pages=1519-21&rft.date=1987-06-19&rft_id=info%3Apmid%2F17835731&rft_id=info%3Adoi%2F10.1126%2Fscience.236.4808.1519&rft_id=info%3Abibcode%2F1987Sci...236.1519K&rft.aulast=Kolata&rft.aufirst=Gina&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASrinivasa+Ramanujan" class="Z3988"></li> <li id="cite_note-Janaki-43">^ <a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Janaki_43-0">a</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Janaki_43-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://web.archive.org/web/20121224003120/https://www.imsc.res.in/~rao/ramanujan/newnow/janaki.pdf">"Ramanujan's wife: Janakiammal (Janaki)"</a> (PDF). Chennai: Institute of Mathematical Sciences. Archived from <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://www.imsc.res.in/~rao/ramanujan/newnow/janaki.pdf">the original</a> (PDF) on 24 December 2012. Retrieved 10 November 2012.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Ramanujan%27s+wife%3A+Janakiammal+%28Janaki%29&rft.place=Chennai&rft.pub=Institute+of+Mathematical+Sciences&rft_id=https%3A%2F%2Fwww.imsc.res.in%2F~rao%2Framanujan%2Fnewnow%2Fjanaki.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASrinivasa+Ramanujan" class="Z3988"></li> <li id="cite_note-express-44">^ <a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-express_44-0">a</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-express_44-1">b</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-express_44-2">c</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJanardhanan2015" class="citation news cs1">Janardhanan, Arun (6 December 2015). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://indianexpress.com/article/lifestyle/life-style/a-passage-to-infinity-the-untold-story-of-srinivasa-ramanujan/">"A passage to infinity"</a>. Indian Express. <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://web.archive.org/web/20160905150332/https://indianexpress.com/article/lifestyle/life-style/a-passage-to-infinity-the-untold-story-of-srinivasa-ramanujan/">Archived</a> from the original on 5 September 2016. Retrieved 7 September 2016.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Indian+Express&rft.atitle=A+passage+to+infinity&rft.date=2015-12-06&rft.aulast=Janardhanan&rft.aufirst=Arun&rft_id=https%3A%2F%2Findianexpress.com%2Farticle%2Flifestyle%2Flife-style%2Fa-passage-to-infinity-the-untold-story-of-srinivasa-ramanujan%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASrinivasa+Ramanujan" class="Z3988"></li> <li id="cite_note-Kanigel-1991-72-45"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Kanigel-1991-72_45-0">^</a> ><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#CITEREFKanigel1991">Kanigel 1991</a>, p. 72</li> <li id="cite_note-46"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-46">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRamanujan1968" class="citation book cs1">Ramanujan, Srinivasa (1968). P. K. Srinivasan (ed.). Ramanujan Memorial Number: Letters and Reminiscences. Vol. 1. Madras: Muthialpet High School. 100.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Ramanujan+Memorial+Number%3A+Letters+and+Reminiscences&rft.place=Madras&rft.pages=100&rft.pub=Muthialpet+High+School&rft.date=1968&rft.aulast=Ramanujan&rft.aufirst=Srinivasa&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASrinivasa+Ramanujan" class="Z3988"></li> <li id="cite_note-47"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-47">^</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#CITEREFKanigel1991">Kanigel 1991</a>, p. 73</li> <li id="cite_note-48"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-48">^</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#CITEREFKanigel1991">Kanigel 1991</a>, pp. 74–75</li> <li id="cite_note-49"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-49">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRanganathan1967" class="citation book cs1"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Shiyali_Ramamrita_Ranganathan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Shiyali Ramamrita Ranganathan">Ranganathan, Shiyali Ramamrita</a> (1967). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://books.google.com/books?id%3DOuTuAAAAMAAJ">Ramanujan: The Man and the Mathematician</a>. Bombay: Asia Publishing House. p. 23. <a href="https://en-m-wikipedia-org.translate.goog/wiki/ISBN_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Special:BookSources/9788185273372?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Special:BookSources/9788185273372"><bdi>9788185273372</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Ramanujan%3A+The+Man+and+the+Mathematician&rft.place=Bombay&rft.pages=23&rft.pub=Asia+Publishing+House&rft.date=1967&rft.isbn=9788185273372&rft.aulast=Ranganathan&rft.aufirst=Shiyali+Ramamrita&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DOuTuAAAAMAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASrinivasa+Ramanujan" class="Z3988"></li> <li id="cite_note-50"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-50">^</a> Srinivasan (1968), Vol. 1, p. 99.</li> <li id="cite_note-rao-51"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-rao_51-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRao,_K._Srinivasa" class="citation web cs1">Rao, K. Srinivasa. <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://web.archive.org/web/20170110132908/https://www.imsc.res.in/~rao/ramanujan/newnow/janaki.pdf">"Ramanujan's wife Janakiammal (Janaki)"</a> (PDF). IMSC. Institute of Mathematical Sciences, Chennai. Archived from <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://www.imsc.res.in/~rao/ramanujan/newnow/janaki.pdf">the original</a> (PDF) on 10 January 2017. Retrieved 7 September 2016.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=IMSC&rft.atitle=Ramanujan%27s+wife+Janakiammal+%28Janaki%29&rft.au=Rao%2C+K.+Srinivasa&rft_id=https%3A%2F%2Fwww.imsc.res.in%2F~rao%2Framanujan%2Fnewnow%2Fjanaki.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASrinivasa+Ramanujan" class="Z3988"></li> <li id="cite_note-52"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-52">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://web.archive.org/web/20161006210739/https://ramanujaninstitute.com/about.html">"About Ramanujan"</a>. The Ramanujan Institute. Archived from <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ramanujaninstitute.com/about.html">the original</a> on 6 October 2016. Retrieved 7 September 2016.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=About+Ramanujan&rft.pub=The+Ramanujan+Institute&rft_id=https%3A%2F%2Framanujaninstitute.com%2Fabout.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASrinivasa+Ramanujan" class="Z3988"></li> <li id="cite_note-Kanigel-1991-77-53">^ <a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Kanigel-1991-77_53-0">a</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Kanigel-1991-77_53-1">b</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#CITEREFKanigel1991">Kanigel 1991</a>, p. 77</li> <li id="cite_note-54"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-54">^</a> 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Res., Bombay. pp. 47–55. <a href="https://en-m-wikipedia-org.translate.goog/wiki/MR_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://mathscinet.ams.org/mathscinet-getitem?mr%3D1441326">1441326</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.atitle=The+circle+method+and+the+Fourier+coefficients+of+modular+forms&rft.btitle=Number+theory+and+related+topics+%28Bombay%2C+1988%29&rft.series=Tata+Inst.+Fund.+Res.+Stud.+Math.&rft.pages=47-55&rft.pub=Tata+Inst.+Fund.+Res.%2C+Bombay&rft.date=1989&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1441326%23id-name%3DMR&rft.aulast=Iwaniec&rft.aufirst=Henryk&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASrinivasa+Ramanujan" class="Z3988"></li> <li id="cite_note-Fox-121">^ <a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Fox_121-0">a</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Fox_121-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation news cs1"><a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://www.foxnews.com/science/100-year-old-deathbed-dreams-of-mathematician-proved-true/">"100-Year-Old Deathbed Dreams of Mathematician Proved True"</a>. Fox News. 28 December 2012. <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://web.archive.org/web/20130107003536/https://www.foxnews.com/science/2012/12/28/mathematician-century-old-secrets-unlocked/">Archived</a> from the original on 7 January 2013.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Fox+News&rft.atitle=100-Year-Old+Deathbed+Dreams+of+Mathematician+Proved+True&rft.date=2012-12-28&rft_id=https%3A%2F%2Fwww.foxnews.com%2Fscience%2F100-year-old-deathbed-dreams-of-mathematician-proved-true%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASrinivasa+Ramanujan" class="Z3988"></li> <li id="cite_note-122"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-122">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWinnie2019" class="citation journal cs1">Winnie, Wen-Ching (9 December 2019). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6939229">"The Ramanujan conjecture and its applications"</a>. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences. 378 (2163). <a href="https://en-m-wikipedia-org.translate.goog/wiki/Doi_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://doi.org/10.1098%252Frsta.2018.0441">10.1098/rsta.2018.0441</a>. <a href="https://en-m-wikipedia-org.translate.goog/wiki/PMC_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="PMC (identifier)">PMC</a> <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6939229">6939229</a>. <a href="https://en-m-wikipedia-org.translate.goog/wiki/PMID_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://pubmed.ncbi.nlm.nih.gov/31813366">31813366</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Philosophical+Transactions+of+the+Royal+Society+A%3A+Mathematical%2C+Physical+and+Engineering+Sciences&rft.atitle=The+Ramanujan+conjecture+and+its+applications&rft.volume=378&rft.issue=2163&rft.date=2019-12-09&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC6939229%23id-name%3DPMC&rft_id=info%3Apmid%2F31813366&rft_id=info%3Adoi%2F10.1098%2Frsta.2018.0441&rft.aulast=Winnie&rft.aufirst=Wen-Ching&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC6939229&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASrinivasa+Ramanujan" class="Z3988"></li> <li id="cite_note-123"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-123">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRamanujan1916" class="citation journal cs1">Ramanujan, Srinivasa (1916). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://web.archive.org/web/20160611175341/https://ramanujan.sirinudi.org/Volumes/published/ram18.pdf">"On certain arithmetical functions"</a> (PDF). Transactions of the Cambridge Philosophical Society. XXII (9). Archived from <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ramanujan.sirinudi.org/Volumes/published/ram18.pdf">the original</a> (PDF) on 11 June 2016. Retrieved 15 May 2016.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Transactions+of+the+Cambridge+Philosophical+Society&rft.atitle=On+certain+arithmetical+functions&rft.volume=XXII&rft.issue=9&rft.date=1916&rft.aulast=Ramanujan&rft.aufirst=Srinivasa&rft_id=https%3A%2F%2Framanujan.sirinudi.org%2FVolumes%2Fpublished%2Fram18.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASrinivasa+Ramanujan" class="Z3988"> The tau function is discussed in pages 194–197.</li> <li id="cite_note-124"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-124">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFOnoAczel2016" class="citation book cs1"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Ken_Ono?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Ken Ono">Ono, Ken</a>; <a href="https://en-m-wikipedia-org.translate.goog/wiki/Amir_Aczel?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Amir Aczel">Aczel, Amir D.</a> (13 April 2016). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://books.google.com/books?id%3DJZkFDAAAQBAJ%26q%3D%2522ideas%2Bthat%2Bwere%2Bcritical%2Bto%2Bthe%2Bproof%2Bof%2BFermat%2527s%2Blast%2Btheorem%2522%26pg%3DPA236">My Search for Ramanujan: How I Learned to Count</a>. <a href="https://en-m-wikipedia-org.translate.goog/wiki/Springer_Science%2BBusiness_Media?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Springer Science+Business Media">Springer</a>. pp. 236–237. <a href="https://en-m-wikipedia-org.translate.goog/wiki/ISBN_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Special:BookSources/978-3319255668?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Special:BookSources/978-3319255668"><bdi>978-3319255668</bdi></a>. <q>ideas that were critical to the proof of Fermat's last theorem</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=My+Search+for+Ramanujan%3A+How+I+Learned+to+Count&rft.pages=236-237&rft.pub=Springer&rft.date=2016-04-13&rft.isbn=978-3319255668&rft.aulast=Ono&rft.aufirst=Ken&rft.au=Aczel%2C+Amir+D.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DJZkFDAAAQBAJ%26q%3D%2522ideas%2Bthat%2Bwere%2Bcritical%2Bto%2Bthe%2Bproof%2Bof%2BFermat%2527s%2Blast%2Btheorem%2522%26pg%3DPA236&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASrinivasa+Ramanujan" class="Z3988"></li> <li id="cite_note-Bruce_Berndt_on_Ramanujan-125">^ <a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Bruce_Berndt_on_Ramanujan_125-0">a</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Bruce_Berndt_on_Ramanujan_125-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBerndt1997" class="citation book cs1">Berndt, Bruce C. (12 December 1997). Ramanujans Notebooks. Springer. <a href="https://en-m-wikipedia-org.translate.goog/wiki/ISBN_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Special:BookSources/978-0387949413?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Special:BookSources/978-0387949413"><bdi>978-0387949413</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Ramanujans+Notebooks&rft.pub=Springer&rft.date=1997-12-12&rft.isbn=978-0387949413&rft.aulast=Berndt&rft.aufirst=Bruce+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASrinivasa+Ramanujan" class="Z3988"></li> <li id="cite_note-126"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-126">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://web.archive.org/web/20120716185939/https://www-gap.dcs.st-and.ac.uk/~history/Quotations/Hardy.html">"Quotations by Hardy"</a>. 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H. <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://www.numbertheory.org/obituaries/LMS/ramanujan/page18.html">"Obituary Notices: Srinivasa Ramanujan"</a>. Proceedings of the London Mathematical Society. 19: lvii. <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://web.archive.org/web/20160305005434/https://www.numbertheory.org/obituaries/LMS/ramanujan/page18.html">Archived</a> from the original on 5 March 2016.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+London+Mathematical+Society&rft.atitle=Obituary+Notices%3A+Srinivasa+Ramanujan&rft.volume=19&rft.pages=lvii&rft.aulast=Hardy&rft.aufirst=G.+H.&rft_id=https%3A%2F%2Fwww.numbertheory.org%2Fobituaries%2FLMS%2Framanujan%2Fpage18.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASrinivasa+Ramanujan" class="Z3988"></li> <li id="cite_note-Ross_2017_u486-128"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Ross_2017_u486_128-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRoss2017" class="citation web cs1">Ross, Greg (6 February 2017). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://www.futilitycloset.com/2017/02/06/podcast-episode-140-ramanujan/">"Podcast Episode 140: Ramanujan"</a>. Futility Closet. Retrieved 24 December 2023.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Futility+Closet&rft.atitle=Podcast+Episode+140%3A+Ramanujan&rft.date=2017-02-06&rft.aulast=Ross&rft.aufirst=Greg&rft_id=https%3A%2F%2Fwww.futilitycloset.com%2F2017%2F02%2F06%2Fpodcast-episode-140-ramanujan%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASrinivasa+Ramanujan" class="Z3988"></li> <li id="cite_note-129"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-129">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJames_R._Newman2000" class="citation book cs1">James R. Newman, ed. (2000). The world of mathematics. 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Retrieved 2 December 2010.</li> <li id="cite_note-alyangar1995-132"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-alyangar1995_132-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAlyangar1995" class="citation book cs1">Alyangar, Srinivasa Ramanujan (1995). Berndt, Bruce C.; Rankin, Robert Alexander (eds.). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://books.google.com/books?id%3DOf5G0r6DQiEC%26pg%3DPA51">Ramanujan: Letters and Commentary</a>. History of Mathematics. Vol. 9. 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Archived from <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://www.imsc.res.in/~rao/ramanujan.html">the original</a> on 16 April 2012. 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T.</a> (1928). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://doi.org/10.1090%252FS0002-9904-1928-04651-7">"Collected Papers of Srinivasa Ramanujan edited by G. H. Hardy, P. V. Seshu Aiyar and B. M. Wilson"</a>. Bulletin of the American Mathematical Society. 34 (6): 783–784. <a href="https://en-m-wikipedia-org.translate.goog/wiki/Doi_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://doi.org/10.1090%252FS0002-9904-1928-04651-7">10.1090/S0002-9904-1928-04651-7</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bulletin+of+the+American+Mathematical+Society&rft.atitle=Collected+Papers+of+Srinivasa+Ramanujan+edited+by+G.+H.+Hardy%2C+P.+V.+Seshu+Aiyar+and+B.+M.+Wilson&rft.volume=34&rft.issue=6&rft.pages=783-784&rft.date=1928&rft_id=info%3Adoi%2F10.1090%2FS0002-9904-1928-04651-7&rft.au=Bell%2C+E.+T.&rft_id=https%3A%2F%2Fdoi.org%2F10.1090%252FS0002-9904-1928-04651-7&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASrinivasa+Ramanujan" class="Z3988"></li> </ol> </div> </div> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(19)"> <h2 id="External_links">External links</h2> </div> <section class="mf-section-19 collapsible-block" id="mf-section-19"> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style> <style data-mw-deduplicate="TemplateStyles:r1250146164">.mw-parser-output .sister-box .side-box-abovebelow{padding:0.75em 0;text-align:center}.mw-parser-output .sister-box .side-box-abovebelow>b{display:block}.mw-parser-output .sister-box .side-box-text>ul{border-top:1px solid #aaa;padding:0.75em 0;width:217px;margin:0 auto}.mw-parser-output .sister-box .side-box-text>ul>li{min-height:31px}.mw-parser-output .sister-logo{display:inline-block;width:31px;line-height:31px;vertical-align:middle;text-align:center}.mw-parser-output .sister-link{display:inline-block;margin-left:4px;width:182px;vertical-align:middle}@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-v2.svg"]{background-color:white}}</style> <div role="navigation" aria-labelledby="sister-projects" class="side-box metadata side-box-right sister-box sistersitebox plainlinks"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"> <div class="side-box-abovebelow">Srinivasa Ramanujan at Wikipedia's <a href="https://en-m-wikipedia-org.translate.goog/wiki/Wikipedia:Wikimedia_sister_projects?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Wikipedia:Wikimedia sister projects">sister projects</a> </div> <div class="side-box-flex"> <div class="side-box-text plainlist"> <ul> <li> <noscript> <img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/20px-Commons-logo.svg.png" decoding="async" width="20" height="27" class="mw-file-element" data-file-width="1024" data-file-height="1376"> </noscript><span class="lazy-image-placeholder" style="width: 20px;height: 27px;" data-src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/20px-Commons-logo.svg.png" data-alt="" data-width="20" data-height="27" data-srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/40px-Commons-logo.svg.png 2x" data-class="mw-file-element"> <a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://commons.wikimedia.org/wiki/Srinivasa_Ramanujan" class="extiw" title="c:Srinivasa Ramanujan">Media</a> from Commons</li> <li> <noscript> <img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/23px-Wikiquote-logo.svg.png" decoding="async" width="23" height="27" class="mw-file-element" data-file-width="300" data-file-height="355"> </noscript><span class="lazy-image-placeholder" style="width: 23px;height: 27px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/23px-Wikiquote-logo.svg.png" data-alt="" data-width="23" data-height="27" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/35px-Wikiquote-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/46px-Wikiquote-logo.svg.png 2x" data-class="mw-file-element"> <a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://en.wikiquote.org/wiki/Srinivasa_Ramanujan" class="extiw" title="q:Srinivasa Ramanujan">Quotations</a> from Wikiquote</li> <li> <noscript> <img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/26px-Wikisource-logo.svg.png" decoding="async" width="26" height="27" class="mw-file-element" data-file-width="410" data-file-height="430"> </noscript><span class="lazy-image-placeholder" style="width: 26px;height: 27px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/26px-Wikisource-logo.svg.png" data-alt="" data-width="26" data-height="27" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/39px-Wikisource-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/51px-Wikisource-logo.svg.png 2x" data-class="mw-file-element"> <a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://en.wikisource.org/wiki/Author:Srinivasa_Ramanujan" class="extiw" title="s:Author:Srinivasa Ramanujan">Texts</a> from Wikisource</li> </ul> </div> </div> </div> <div class="mw-heading mw-heading3"> <h3 id="Media_links">Media links</h3> </div> <ul> <li> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBiswas2006" class="citation news cs1">Biswas, Soutik (16 March 2006). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://news.bbc.co.uk/2/hi/south_asia/4811920.stm">"Film to celebrate mathematics genius"</a>. BBC. Retrieved 24 August 2006.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=BBC&rft.atitle=Film+to+celebrate+mathematics+genius&rft.date=2006-03-16&rft.aulast=Biswas&rft.aufirst=Soutik&rft_id=https%3A%2F%2Fnews.bbc.co.uk%2F2%2Fhi%2Fsouth_asia%2F4811920.stm&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASrinivasa+Ramanujan" class="Z3988"></li> <li><a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://devbenegal.com/2006/03/15/feature-film-on-math-genius-ramanujan/">Feature Film on Mathematics Genius Ramanujan by Dev Benegal and Stephen Fry</a></li> <li><a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://www.bbc.co.uk/radio4/science/further5.shtml">BBC radio programme about Ramanujan – episode 5</a></li> <li><a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://archive.org/details/Ramanujan">A biographical song about Ramanujan's life</a></li> <li> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://www.youtube.com/watch?v%3Dixra6x1XdjE">"Why Did This Mathematician's Equations Make Everyone So Angry?"</a>. Youtube.com. Thoughty2. 11 April 2022. Retrieved 29 June 2022.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Youtube.com&rft.atitle=Why+Did+This+Mathematician%27s+Equations+Make+Everyone+So+Angry%3F&rft.date=2022-04-11&rft_id=https%3A%2F%2Fwww.youtube.com%2Fwatch%3Fv%3Dixra6x1XdjE&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASrinivasa+Ramanujan" class="Z3988"></li> </ul> <div class="mw-heading mw-heading3"> <h3 id="Biographical_links">Biographical links</h3> </div> <ul> <li><a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://mathgenealogy.org/id.php?id%3D91561">Srinivasa Ramanujan</a> at the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Mathematics_Genealogy_Project?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Mathematics Genealogy Project">Mathematics Genealogy Project</a></li> <li> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFO'ConnorRobertson" class="citation cs2">O'Connor, John J.; <a href="https://en-m-wikipedia-org.translate.goog/wiki/Edmund_F._Robertson?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edmund F. Robertson">Robertson, Edmund F.</a>, <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://mathshistory.st-andrews.ac.uk/Biographies/Ramanujan.html">"Srinivasa Ramanujan"</a>, <a href="https://en-m-wikipedia-org.translate.goog/wiki/MacTutor_History_of_Mathematics_Archive?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="MacTutor History of Mathematics Archive">MacTutor History of Mathematics Archive</a>, <a href="https://en-m-wikipedia-org.translate.goog/wiki/University_of_St_Andrews?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="University of St Andrews">University of St Andrews</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Srinivasa+Ramanujan&rft.btitle=MacTutor+History+of+Mathematics+Archive&rft.pub=University+of+St+Andrews&rft.aulast=O%27Connor&rft.aufirst=John+J.&rft.au=Robertson%2C+Edmund+F.&rft_id=https%3A%2F%2Fmathshistory.st-andrews.ac.uk%2FBiographies%2FRamanujan.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASrinivasa+Ramanujan" class="Z3988"></li> <li> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Eric_Wolfgang_Weisstein?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Eric Wolfgang Weisstein">Weisstein, Eric Wolfgang</a> (ed.). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=http://scienceworld.wolfram.com/biography/Ramanujan.html">"Ramanujan, Srinivasa (1887–1920)"</a>. <a href="https://en-m-wikipedia-org.translate.goog/wiki/ScienceWorld?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="ScienceWorld">ScienceWorld</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=ScienceWorld&rft.atitle=Ramanujan%2C+Srinivasa+%281887%E2%80%931920%29&rft_id=http%3A%2F%2Fscienceworld.wolfram.com%2Fbiography%2FRamanujan.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASrinivasa+Ramanujan" class="Z3988"></li> <li><a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://www.usna.edu/Users/math/meh/ramanujan.html">A short biography of Ramanujan</a></li> <li><a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://web.archive.org/web/20180805161945/http://srinivasaramanujan.in/">"Our Devoted Site for Great Mathematical Genius"</a></li> </ul> <div class="mw-heading mw-heading3"> <h3 id="Other_links">Other links</h3> </div> <ul> <li> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWolfram2016" class="citation web cs1"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Stephen_Wolfram?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Stephen Wolfram">Wolfram, Stephen</a> (27 April 2016). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://writings.stephenwolfram.com/2016/04/who-was-ramanujan/">"Who Was Ramanujan?"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Who+Was+Ramanujan%3F&rft.date=2016-04-27&rft.aulast=Wolfram&rft.aufirst=Stephen&rft_id=https%3A%2F%2Fwritings.stephenwolfram.com%2F2016%2F04%2Fwho-was-ramanujan%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASrinivasa+Ramanujan" class="Z3988"></li> <li>A Study Group For Mathematics: <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://archive.today/20121204163504/http://groups.yahoo.com/group/srinivasaramanujan/">Srinivasa Ramanujan Iyengar</a></li> <li><a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://web.archive.org/web/20020126193110/https://www.math.ufl.edu/~frank/ramanujan.html">The Ramanujan Journal</a> – An international journal devoted to Ramanujan</li> <li><a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://web.archive.org/web/20051125142618/https://www.mathunion.org/General/Prizes/">International Math Union Prizes</a>, including a Ramanujan Prize</li> <li>Hindu.com: <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://web.archive.org/web/20050120000345/http://www.hindu.com/mag/2004/12/26/stories/2004122600610400.htm">Norwegian and Indian mathematical geniuses</a>, <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://www.hindu.com/thehindu/br/2003/08/26/stories/2003082600120300.htm">Ramanujan – Essays and Surveys</a> <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://web.archive.org/web/20121106022700/http://www.hindu.com/thehindu/br/2003/08/26/stories/2003082600120300.htm">Archived</a> 6 November 2012 at the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Wayback_Machine?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Wayback Machine">Wayback Machine</a>, <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://web.archive.org/web/20040103074953/http://www.hindu.com/2003/12/22/stories/2003122204061100.htm">Ramanujan's growing influence</a>, <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://web.archive.org/web/20040628115315/http://www.hindu.com/thehindu/mag/2002/12/22/stories/2002122200040400.htm">Ramanujan's mentor</a></li> <li>Hindu.com: <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://www.thehindu.com/life-and-style/metroplus/article884010.ece">The sponsor of Ramanujan</a></li> <li> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBruce_C._BerndtRobert_A._Rankin2000" class="citation journal cs1">Bruce C. Berndt; Robert A. Rankin (2000). "The Books Studied by Ramanujan in India". <a href="https://en-m-wikipedia-org.translate.goog/wiki/American_Mathematical_Monthly?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="American Mathematical Monthly">American Mathematical Monthly</a>. 107 (7): 595–601. <a href="https://en-m-wikipedia-org.translate.goog/wiki/Doi_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://doi.org/10.2307%252F2589114">10.2307/2589114</a>. <a href="https://en-m-wikipedia-org.translate.goog/wiki/JSTOR_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://www.jstor.org/stable/2589114">2589114</a>. <a href="https://en-m-wikipedia-org.translate.goog/wiki/MR_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://mathscinet.ams.org/mathscinet-getitem?mr%3D1786233">1786233</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Mathematical+Monthly&rft.atitle=The+Books+Studied+by+Ramanujan+in+India&rft.volume=107&rft.issue=7&rft.pages=595-601&rft.date=2000&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1786233%23id-name%3DMR&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2589114%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F2589114&rft.au=Bruce+C.+Berndt&rft.au=Robert+A.+Rankin&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASrinivasa+Ramanujan" class="Z3988"></li> <li><a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://www.maa.org/news/030807puzzlesolved.html">"Ramanujan's mock theta function puzzle solved"</a></li> <li><a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://www.imsc.res.in/~rao/ramanujan/contentindex.html">Ramanujan's papers and notebooks</a></li> <li><a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=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</a></li> <li><a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://web.archive.org/web/20120319203227/http://www.friedeye.com/2011/01/01/ramanujan/">Ramanujan</a> on Fried Eye</li> <li> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFClark" class="citation web cs1">Clark, Alex. <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://web.archive.org/web/20180204091812/http://www.numberphile.com/videos/163.html">"163 and Ramanujan Constant"</a>. Numberphile. <a href="https://en-m-wikipedia-org.translate.goog/wiki/Brady_Haran?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Brady Haran">Brady Haran</a>. Archived from <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://www.numberphile.com/videos/163.html">the original</a> on 4 February 2018. 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<li class="interlanguage-link interwiki-awa mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://awa.wikipedia.org/wiki/%25E0%25A4%25B6%25E0%25A5%258D%25E0%25A4%25B0%25E0%25A5%2580%25E0%25A4%25A8%25E0%25A4%25BF%25E0%25A4%25B5%25E0%25A4%25BE%25E0%25A4%25B8_%25E0%25A4%25B0%25E0%25A4%25BE%25E0%25A4%25AE%25E0%25A4%25BE%25E0%25A4%25A8%25E0%25A5%2581%25E0%25A4%259C%25E0%25A4%25A8%25E0%25A5%258D" title="श्रीनिवास रामानुजन् – Awadhi" lang="awa" hreflang="awa" data-title="श्रीनिवास रामानुजन्" data-language-autonym="अवधी" data-language-local-name="Awadhi" class="interlanguage-link-target">अवधी</a></li> <li class="interlanguage-link interwiki-az mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://az.wikipedia.org/wiki/Srinivasa_Ramanucan" title="Srinivasa Ramanucan – Azerbaijani" lang="az" hreflang="az" data-title="Srinivasa Ramanucan" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target">Azərbaycanca</a></li> <li class="interlanguage-link interwiki-azb mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://azb.wikipedia.org/wiki/%25D8%25B3%25D8%25B1%25DB%258C%25D9%2586%25DB%258C%25D9%2588%25D8%25A7%25D8%25B3%25D8%25A7_%25D8%25B1%25D8%25A7%25D9%2585%25D8%25A7%25D9%2586%25D9%2588%25D8%25AC%25D8%25A7%25D9%2586" title="سرینیواسا رامانوجان – South Azerbaijani" lang="azb" hreflang="azb" data-title="سرینیواسا رامانوجان" data-language-autonym="تۆرکجه" data-language-local-name="South Azerbaijani" class="interlanguage-link-target">تۆرکجه</a></li> <li class="interlanguage-link interwiki-bn mw-list-item"><a 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Рамануджан" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target">Беларуская</a></li> <li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://bg.wikipedia.org/wiki/%25D0%25A1%25D1%2580%25D0%25B8%25D0%25BD%25D0%25B8%25D0%25B2%25D0%25B0%25D1%2581%25D0%25B0_%25D0%25A0%25D0%25B0%25D0%25BC%25D0%25B0%25D0%25BD%25D1%2583%25D0%25B4%25D0%25B6%25D0%25B0%25D0%25BD" title="Сриниваса Рамануджан – Bulgarian" lang="bg" hreflang="bg" data-title="Сриниваса Рамануджан" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target">Български</a></li> <li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://bs.wikipedia.org/wiki/Srinivasa_Aaiyangar_Ramanujan" title="Srinivasa Aaiyangar Ramanujan – Bosnian" lang="bs" hreflang="bs" data-title="Srinivasa Aaiyangar Ramanujan" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target">Bosanski</a></li> <li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ca.wikipedia.org/wiki/Srinivasa_Ramanujan" title="Srinivasa Ramanujan – Catalan" lang="ca" hreflang="ca" data-title="Srinivasa Ramanujan" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li> <li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://cv.wikipedia.org/wiki/%25D0%25A1%25D1%2580%25D0%25B8%25D0%25BD%25D0%25B8%25D0%25B2%25D0%25B0%25D1%2581%25D0%25B0_%25D0%25A0%25D0%25B0%25D0%25BC%25D0%25B0%25D0%25BD%25D1%2583%25D0%25B4%25D0%25B6%25D0%25B0%25D0%25BD_%25D0%2590%25D0%25B9%25D0%25B5%25D0%25BD%25D0%25B3%25D0%25BE%25D1%2580" title="Сриниваса Рамануджан Айенгор – Chuvash" lang="cv" hreflang="cv" data-title="Сриниваса Рамануджан Айенгор" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target">Чӑвашла</a></li> <li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://cs.wikipedia.org/wiki/%25C5%25A0r%25C3%25ADniv%25C3%25A1sa_R%25C3%25A1manud%25C5%25BEan" title="Šrínivása Rámanudžan – Czech" lang="cs" hreflang="cs" data-title="Šrínivása Rámanudžan" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li> <li class="interlanguage-link interwiki-da mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://da.wikipedia.org/wiki/Srinivasa_Aiyangar_Ramanujan" title="Srinivasa Aiyangar Ramanujan – Danish" lang="da" hreflang="da" data-title="Srinivasa Aiyangar Ramanujan" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target">Dansk</a></li> <li class="interlanguage-link interwiki-de badge-Q17437798 badge-goodarticle mw-list-item" title="good article badge"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://de.wikipedia.org/wiki/Srinivasa_Ramanujan" title="Srinivasa Ramanujan – German" lang="de" hreflang="de" data-title="Srinivasa Ramanujan" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li> <li class="interlanguage-link interwiki-et mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://et.wikipedia.org/wiki/Srinivasa_Ramanujan" title="Srinivasa Ramanujan – Estonian" lang="et" hreflang="et" data-title="Srinivasa Ramanujan" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target">Eesti</a></li> <li class="interlanguage-link interwiki-el mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://el.wikipedia.org/wiki/%25CE%25A3%25CF%2581%25CE%25B9%25CE%25BD%25CE%25B9%25CE%25B2%25CE%25AC%25CF%2583%25CE%25B1_%25CE%25A1%25CE%25B1%25CE%25BC%25CE%25B1%25CE%25BD%25CE%25BF%25CF%258D%25CF%2584%25CE%25B6%25CE%25B1%25CE%25BD" title="Σρινιβάσα Ραμανούτζαν – Greek" lang="el" hreflang="el" data-title="Σρινιβάσα Ραμανούτζαν" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target">Ελληνικά</a></li> <li class="interlanguage-link interwiki-es mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://es.wikipedia.org/wiki/Srinivasa_Ramanujan" title="Srinivasa Ramanujan – Spanish" lang="es" hreflang="es" data-title="Srinivasa Ramanujan" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li> <li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://eo.wikipedia.org/wiki/Srinivasa_Aiyangar_Ramanujan" title="Srinivasa Aiyangar Ramanujan – Esperanto" lang="eo" hreflang="eo" data-title="Srinivasa Aiyangar Ramanujan" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target">Esperanto</a></li> <li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://eu.wikipedia.org/wiki/Srinivasa_Aiyangar_Ramanujan" title="Srinivasa Aiyangar Ramanujan – Basque" lang="eu" hreflang="eu" data-title="Srinivasa Aiyangar Ramanujan" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target">Euskara</a></li> <li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://fa.wikipedia.org/wiki/%25D8%25B3%25D8%25B1%25DB%258C%25D9%2586%25DB%258C%25D9%2588%25D8%25A7%25D8%25B3%25D8%25A7_%25D8%25B1%25D8%25A7%25D9%2585%25D8%25A7%25D9%2586%25D9%2588%25D8%25AC%25D8%25A7%25D9%2586" title="سرینیواسا رامانوجان – Persian" lang="fa" hreflang="fa" data-title="سرینیواسا رامانوجان" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li> <li class="interlanguage-link interwiki-hif mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://hif.wikipedia.org/wiki/Srinivasa_Ramanujan" title="Srinivasa Ramanujan – Fiji Hindi" lang="hif" hreflang="hif" data-title="Srinivasa Ramanujan" data-language-autonym="Fiji Hindi" data-language-local-name="Fiji Hindi" class="interlanguage-link-target">Fiji Hindi</a></li> <li class="interlanguage-link interwiki-fr badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://fr.wikipedia.org/wiki/Srinivasa_Ramanujan" title="Srinivasa Ramanujan – French" lang="fr" hreflang="fr" data-title="Srinivasa Ramanujan" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li> <li class="interlanguage-link interwiki-fy mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://fy.wikipedia.org/wiki/Srinivasa_Ramanujan" title="Srinivasa Ramanujan – Western Frisian" lang="fy" hreflang="fy" data-title="Srinivasa Ramanujan" data-language-autonym="Frysk" data-language-local-name="Western Frisian" class="interlanguage-link-target">Frysk</a></li> <li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ga.wikipedia.org/wiki/Srinivasa_Ramanujan" title="Srinivasa Ramanujan – Irish" lang="ga" hreflang="ga" data-title="Srinivasa Ramanujan" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target">Gaeilge</a></li> <li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://gl.wikipedia.org/wiki/Srinivasa_Ramanujan" title="Srinivasa Ramanujan – Galician" lang="gl" hreflang="gl" data-title="Srinivasa Ramanujan" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target">Galego</a></li> <li class="interlanguage-link interwiki-gu mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://gu.wikipedia.org/wiki/%25E0%25AA%25B6%25E0%25AB%258D%25E0%25AA%25B0%25E0%25AB%2580%25E0%25AA%25A8%25E0%25AA%25BF%25E0%25AA%25B5%25E0%25AA%25BE%25E0%25AA%25B8_%25E0%25AA%25B0%25E0%25AA%25BE%25E0%25AA%25AE%25E0%25AA%25BE%25E0%25AA%25A8%25E0%25AB%2581%25E0%25AA%259C%25E0%25AA%25A8" title="શ્રીનિવાસ રામાનુજન – Gujarati" lang="gu" hreflang="gu" data-title="શ્રીનિવાસ રામાનુજન" data-language-autonym="ગુજરાતી" data-language-local-name="Gujarati" class="interlanguage-link-target">ગુજરાતી</a></li> <li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ko.wikipedia.org/wiki/%25EC%258A%25A4%25EB%25A6%25AC%25EB%258B%2588%25EB%25B0%2594%25EC%2582%25AC_%25EB%259D%25BC%25EB%25A7%2588%25EB%2588%2584%25EC%259E%2594" title="스리니바사 라마누잔 – Korean" lang="ko" hreflang="ko" data-title="스리니바사 라마누잔" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li> <li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://hy.wikipedia.org/wiki/%25D5%258D%25D6%2580%25D5%25AB%25D5%25B6%25D5%25AB%25D5%25BE%25D5%25A1%25D5%25BD%25D5%25A1_%25D5%258C%25D5%25A1%25D5%25B4%25D5%25A1%25D5%25B6%25D5%25B8%25D6%2582%25D5%25BB%25D5%25A1%25D5%25B6" title="Սրինիվասա Ռամանուջան – Armenian" lang="hy" hreflang="hy" data-title="Սրինիվասա Ռամանուջան" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target">Հայերեն</a></li> <li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://hi.wikipedia.org/wiki/%25E0%25A4%25B6%25E0%25A5%258D%25E0%25A4%25B0%25E0%25A5%2580%25E0%25A4%25A8%25E0%25A4%25BF%25E0%25A4%25B5%25E0%25A4%25BE%25E0%25A4%25B8_%25E0%25A4%25B0%25E0%25A4%25BE%25E0%25A4%25AE%25E0%25A4%25BE%25E0%25A4%25A8%25E0%25A5%2581%25E0%25A4%259C%25E0%25A4%25A8%25E0%25A5%258D" title="श्रीनिवास रामानुजन् – Hindi" lang="hi" hreflang="hi" data-title="श्रीनिवास रामानुजन्" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target">हिन्दी</a></li> <li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://hr.wikipedia.org/wiki/Srinivasa_Ramanujan" title="Srinivasa Ramanujan – Croatian" lang="hr" hreflang="hr" data-title="Srinivasa Ramanujan" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target">Hrvatski</a></li> <li class="interlanguage-link interwiki-io mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://io.wikipedia.org/wiki/Srinivasa_Ramanujan" title="Srinivasa Ramanujan – Ido" lang="io" hreflang="io" data-title="Srinivasa Ramanujan" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target">Ido</a></li> <li class="interlanguage-link interwiki-bpy mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://bpy.wikipedia.org/wiki/%25E0%25A6%25B6%25E0%25A7%258D%25E0%25A6%25B0%25E0%25A7%2580%25E0%25A6%25A8%25E0%25A6%25BF%25E0%25A6%25AC%25E0%25A6%25BE%25E0%25A6%25B8_%25E0%25A6%25B0%25E0%25A6%25BE%25E0%25A6%25AE%25E0%25A6%25BE%25E0%25A6%25A8%25E0%25A7%2581%25E0%25A6%259C%25E0%25A6%25A8" title="শ্রীনিবাস রামানুজন – Bishnupriya" lang="bpy" hreflang="bpy" data-title="শ্রীনিবাস রামানুজন" data-language-autonym="বিষ্ণুপ্রিয়া মণিপুরী" data-language-local-name="Bishnupriya" class="interlanguage-link-target">বিষ্ণুপ্রিয়া মণিপুরী</a></li> <li class="interlanguage-link interwiki-id mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://id.wikipedia.org/wiki/Srinivasa_Ramanujan" title="Srinivasa Ramanujan – Indonesian" lang="id" hreflang="id" data-title="Srinivasa Ramanujan" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target">Bahasa Indonesia</a></li> <li class="interlanguage-link interwiki-is mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://is.wikipedia.org/wiki/Srinivasa_Ramanujan" title="Srinivasa Ramanujan – Icelandic" lang="is" hreflang="is" data-title="Srinivasa Ramanujan" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target">Íslenska</a></li> <li class="interlanguage-link interwiki-it mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://it.wikipedia.org/wiki/Srinivasa_Ramanujan" title="Srinivasa Ramanujan – Italian" lang="it" hreflang="it" data-title="Srinivasa Ramanujan" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li> <li class="interlanguage-link interwiki-he mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://he.wikipedia.org/wiki/%25D7%25A1%25D7%25A8%25D7%2599%25D7%25A0%25D7%2599%25D7%2595%25D7%2595%25D7%2590%25D7%25A1%25D7%2594_%25D7%25A8%25D7%259E%25D7%25A0%25D7%2595%25D7%2592%2527%25D7%2590%25D7%259F" title="סריניוואסה רמנוג'אן – Hebrew" lang="he" hreflang="he" data-title="סריניוואסה רמנוג'אן" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li> <li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://kn.wikipedia.org/wiki/%25E0%25B2%25B6%25E0%25B3%258D%25E0%25B2%25B0%25E0%25B3%2580%25E0%25B2%25A8%25E0%25B2%25BF%25E0%25B2%25B5%25E0%25B2%25BE%25E0%25B2%25B8_%25E0%25B2%25B0%25E0%25B2%25BE%25E0%25B2%25AE%25E0%25B2%25BE%25E0%25B2%25A8%25E0%25B3%2581%25E0%25B2%259C%25E0%25B2%25A8%25E0%25B3%258D" title="ಶ್ರೀನಿವಾಸ ರಾಮಾನುಜನ್ – Kannada" lang="kn" hreflang="kn" data-title="ಶ್ರೀನಿವಾಸ ರಾಮಾನುಜನ್" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Kannada" class="interlanguage-link-target">ಕನ್ನಡ</a></li> <li class="interlanguage-link interwiki-pam mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://pam.wikipedia.org/wiki/Srinivasa_Ramanujan" title="Srinivasa Ramanujan – Pampanga" lang="pam" hreflang="pam" data-title="Srinivasa Ramanujan" data-language-autonym="Kapampangan" data-language-local-name="Pampanga" class="interlanguage-link-target">Kapampangan</a></li> <li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ka.wikipedia.org/wiki/%25E1%2583%25A1%25E1%2583%25A0%25E1%2583%2598%25E1%2583%259C%25E1%2583%2598%25E1%2583%2595%25E1%2583%2590%25E1%2583%25A1%25E1%2583%2590_%25E1%2583%25A0%25E1%2583%2590%25E1%2583%259B%25E1%2583%2590%25E1%2583%259C%25E1%2583%25A3%25E1%2583%25AF%25E1%2583%2590%25E1%2583%259C%25E1%2583%2598" title="სრინივასა რამანუჯანი – Georgian" lang="ka" hreflang="ka" data-title="სრინივასა რამანუჯანი" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target">ქართული</a></li> <li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://kk.wikipedia.org/wiki/%25D0%25A1%25D1%2580%25D0%25B8%25D0%25BD%25D0%25B8%25D0%25B2%25D0%25B0%25D1%2581%25D0%25B0_%25D0%25A0%25D0%25B0%25D0%25BC%25D0%25B0%25D0%25BD%25D1%2583%25D0%25B6%25D0%25B0%25D0%25BD_%25D0%2590%25D0%25B9%25D0%25B5%25D0%25BD%25D0%25B3%25D0%25BE%25D1%2580" title="Сриниваса Раманужан Айенгор – Kazakh" lang="kk" hreflang="kk" data-title="Сриниваса Раманужан Айенгор" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target">Қазақша</a></li> <li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ht.wikipedia.org/wiki/Srinivasa_Ramanujan" title="Srinivasa Ramanujan – Haitian Creole" lang="ht" hreflang="ht" data-title="Srinivasa Ramanujan" data-language-autonym="Kreyòl ayisyen" data-language-local-name="Haitian Creole" class="interlanguage-link-target">Kreyòl ayisyen</a></li> <li class="interlanguage-link interwiki-la mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://la.wikipedia.org/wiki/Srinivasa_Aiyangar_Ramanujan" title="Srinivasa Aiyangar Ramanujan – Latin" lang="la" hreflang="la" data-title="Srinivasa Aiyangar Ramanujan" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target">Latina</a></li> <li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://lv.wikipedia.org/wiki/%25C5%25A0r%25C4%25ABniv%25C4%2581sa_R%25C4%2581m%25C4%2581nud%25C5%25BEans" title="Šrīnivāsa Rāmānudžans – Latvian" lang="lv" hreflang="lv" data-title="Šrīnivāsa Rāmānudžans" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target">Latviešu</a></li> <li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://hu.wikipedia.org/wiki/Sr%25C3%25ADniv%25C3%25A1sza_R%25C3%25A1m%25C3%25A1nudzsan" title="Srínivásza Rámánudzsan – Hungarian" lang="hu" hreflang="hu" data-title="Srínivásza Rámánudzsan" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target">Magyar</a></li> <li class="interlanguage-link interwiki-mg mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://mg.wikipedia.org/wiki/Srinivasa_Ramanujan" title="Srinivasa Ramanujan – Malagasy" lang="mg" hreflang="mg" data-title="Srinivasa Ramanujan" data-language-autonym="Malagasy" data-language-local-name="Malagasy" class="interlanguage-link-target">Malagasy</a></li> <li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ml.wikipedia.org/wiki/%25E0%25B4%25B6%25E0%25B5%258D%25E0%25B4%25B0%25E0%25B5%2580%25E0%25B4%25A8%25E0%25B4%25BF%25E0%25B4%25B5%25E0%25B4%25BE%25E0%25B4%25B8_%25E0%25B4%25B0%25E0%25B4%25BE%25E0%25B4%25AE%25E0%25B4%25BE%25E0%25B4%25A8%25E0%25B5%2581%25E0%25B4%259C%25E0%25B5%25BB" title="ശ്രീനിവാസ രാമാനുജൻ – Malayalam" lang="ml" hreflang="ml" data-title="ശ്രീനിവാസ രാമാനുജൻ" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target">മലയാളം</a></li> <li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://mr.wikipedia.org/wiki/%25E0%25A4%25B6%25E0%25A5%258D%25E0%25A4%25B0%25E0%25A5%2580%25E0%25A4%25A8%25E0%25A4%25BF%25E0%25A4%25B5%25E0%25A4%25BE%25E0%25A4%25B8_%25E0%25A4%25B0%25E0%25A4%25BE%25E0%25A4%25AE%25E0%25A4%25BE%25E0%25A4%25A8%25E0%25A5%2581%25E0%25A4%259C%25E0%25A4%25A8" title="श्रीनिवास रामानुजन – Marathi" lang="mr" hreflang="mr" data-title="श्रीनिवास रामानुजन" data-language-autonym="मराठी" data-language-local-name="Marathi" class="interlanguage-link-target">मराठी</a></li> <li class="interlanguage-link interwiki-xmf mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://xmf.wikipedia.org/wiki/%25E1%2583%25A1%25E1%2583%25A0%25E1%2583%2598%25E1%2583%259C%25E1%2583%2598%25E1%2583%2595%25E1%2583%2590%25E1%2583%25A1%25E1%2583%2590_%25E1%2583%25A0%25E1%2583%2590%25E1%2583%259B%25E1%2583%2590%25E1%2583%259C%25E1%2583%25A3%25E1%2583%25AF%25E1%2583%2590%25E1%2583%259C%25E1%2583%2598" title="სრინივასა რამანუჯანი – Mingrelian" lang="xmf" hreflang="xmf" data-title="სრინივასა რამანუჯანი" data-language-autonym="მარგალური" data-language-local-name="Mingrelian" class="interlanguage-link-target">მარგალური</a></li> <li class="interlanguage-link interwiki-arz mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://arz.wikipedia.org/wiki/%25D8%25B3%25D8%25B1%25D9%258A%25D9%2586%25D9%2581%25D8%25A7%25D8%25B3%25D8%25A7_%25D8%25A7%25D9%258A%25D9%2586%25D8%25AC%25D8%25A7%25D8%25B1_%25D8%25B1%25D8%25A7%25D9%2585%25D8%25A7%25D9%2586%25D8%25AC%25D9%2586" title="سرينفاسا اينجار رامانجن – Egyptian Arabic" lang="arz" hreflang="arz" data-title="سرينفاسا اينجار رامانجن" data-language-autonym="مصرى" data-language-local-name="Egyptian Arabic" class="interlanguage-link-target">مصرى</a></li> <li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ms.wikipedia.org/wiki/Srinivasa_Ramanujan" title="Srinivasa Ramanujan – Malay" lang="ms" hreflang="ms" data-title="Srinivasa Ramanujan" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target">Bahasa Melayu</a></li> <li class="interlanguage-link interwiki-mwl mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://mwl.wikipedia.org/wiki/Srinivasa_Ramanujan" title="Srinivasa Ramanujan – Mirandese" lang="mwl" hreflang="mwl" data-title="Srinivasa Ramanujan" data-language-autonym="Mirandés" data-language-local-name="Mirandese" class="interlanguage-link-target">Mirandés</a></li> <li class="interlanguage-link interwiki-my mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://my.wikipedia.org/wiki/%25E1%2580%259B%25E1%2580%25BE%25E1%2580%259B%25E1%2580%25AE%25E1%2580%2594%25E1%2580%25AE%25E1%2580%2597%25E1%2580%25AC%25E1%2580%2586%25E1%2580%25AC_%25E1%2580%259B%25E1%2580%25AC%25E1%2580%2599%25E1%2580%2594%25E1%2580%25BA%25E1%2580%2594%25E1%2580%25BB%25E1%2580%25B0%25E1%2580%2582%25E1%2580%25BB%25E1%2580%2594%25E1%2580%25BA" title="ရှရီနီဗာဆာ ရာမန်နျူဂျန် – Burmese" lang="my" hreflang="my" data-title="ရှရီနီဗာဆာ ရာမန်နျူဂျန်" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="Burmese" class="interlanguage-link-target">မြန်မာဘာသာ</a></li> <li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://nl.wikipedia.org/wiki/Srinivasa_Aaiyangar_Ramanujan" title="Srinivasa Aaiyangar Ramanujan – Dutch" lang="nl" hreflang="nl" data-title="Srinivasa Aaiyangar Ramanujan" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li> <li class="interlanguage-link interwiki-ne mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ne.wikipedia.org/wiki/%25E0%25A4%25B6%25E0%25A5%258D%25E0%25A4%25B0%25E0%25A5%2580%25E0%25A4%25A8%25E0%25A4%25BF%25E0%25A4%25B5%25E0%25A4%25BE%25E0%25A4%25B8_%25E0%25A4%25B0%25E0%25A4%25BE%25E0%25A4%25AE%25E0%25A4%25BE%25E0%25A4%25A8%25E0%25A5%2581%25E0%25A4%259C%25E0%25A4%25A8" title="श्रीनिवास रामानुजन – Nepali" lang="ne" hreflang="ne" data-title="श्रीनिवास रामानुजन" data-language-autonym="नेपाली" data-language-local-name="Nepali" class="interlanguage-link-target">नेपाली</a></li> <li class="interlanguage-link interwiki-new mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://new.wikipedia.org/wiki/%25E0%25A4%25B6%25E0%25A5%258D%25E0%25A4%25B0%25E0%25A5%2580%25E0%25A4%25A8%25E0%25A4%25BF%25E0%25A4%25B5%25E0%25A4%25BE%25E0%25A4%25B8_%25E0%25A4%25B0%25E0%25A4%25BE%25E0%25A4%25AE%25E0%25A4%25BE%25E0%25A4%25A8%25E0%25A5%2581%25E0%25A4%259C%25E0%25A4%25A8%25E0%25A5%258D" title="श्रीनिवास रामानुजन् – Newari" lang="new" hreflang="new" data-title="श्रीनिवास रामानुजन्" data-language-autonym="नेपाल भाषा" data-language-local-name="Newari" class="interlanguage-link-target">नेपाल भाषा</a></li> <li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ja.wikipedia.org/wiki/%25E3%2582%25B7%25E3%2583%25A5%25E3%2583%25AA%25E3%2583%258B%25E3%2583%25B4%25E3%2582%25A1%25E3%2583%25BC%25E3%2582%25B5%25E3%2583%25BB%25E3%2583%25A9%25E3%2583%259E%25E3%2583%258C%25E3%2582%25B8%25E3%2583%25A3%25E3%2583%25B3" title="シュリニヴァーサ・ラマヌジャン – Japanese" lang="ja" hreflang="ja" data-title="シュリニヴァーサ・ラマヌジャン" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li> <li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://frr.wikipedia.org/wiki/Srinivasa_Ramanujan" title="Srinivasa Ramanujan – Northern Frisian" lang="frr" hreflang="frr" data-title="Srinivasa Ramanujan" data-language-autonym="Nordfriisk" data-language-local-name="Northern Frisian" class="interlanguage-link-target">Nordfriisk</a></li> <li class="interlanguage-link interwiki-no mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://no.wikipedia.org/wiki/Srinivasa_Aiyangar_Ramanujan" title="Srinivasa Aiyangar Ramanujan – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Srinivasa Aiyangar Ramanujan" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target">Norsk bokmål</a></li> <li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://nn.wikipedia.org/wiki/Srinivasa_Ramanujan" title="Srinivasa Ramanujan – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Srinivasa Ramanujan" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target">Norsk nynorsk</a></li> <li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://oc.wikipedia.org/wiki/Srinivasa_Ramanujan" title="Srinivasa Ramanujan – Occitan" lang="oc" hreflang="oc" data-title="Srinivasa Ramanujan" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target">Occitan</a></li> <li class="interlanguage-link interwiki-or mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://or.wikipedia.org/wiki/%25E0%25AC%25B6%25E0%25AD%258D%25E0%25AC%25B0%25E0%25AD%2580%25E0%25AC%25A8%25E0%25AC%25BF%25E0%25AC%25AC%25E0%25AC%25BE%25E0%25AC%25B8_%25E0%25AC%25B0%25E0%25AC%25BE%25E0%25AC%25AE%25E0%25AC%25BE%25E0%25AC%25A8%25E0%25AD%2581%25E0%25AC%259C%25E0%25AC%25A8" title="ଶ୍ରୀନିବାସ ରାମାନୁଜନ – Odia" lang="or" hreflang="or" data-title="ଶ୍ରୀନିବାସ ରାମାନୁଜନ" data-language-autonym="ଓଡ଼ିଆ" data-language-local-name="Odia" class="interlanguage-link-target">ଓଡ଼ିଆ</a></li> <li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://pa.wikipedia.org/wiki/%25E0%25A8%25B8%25E0%25A8%25BC%25E0%25A9%258D%25E0%25A8%25B0%25E0%25A9%2580%25E0%25A8%25A8%25E0%25A8%25BF%25E0%25A8%25B5%25E0%25A8%25BE%25E0%25A8%25B8_%25E0%25A8%25B0%25E0%25A8%25BE%25E0%25A8%25AE%25E0%25A8%25BE%25E0%25A8%25A8%25E0%25A9%2581%25E0%25A8%259C%25E0%25A8%25A8_%25E0%25A8%2586%25E0%25A8%2587%25E0%25A9%25B0%25E0%25A8%2597%25E0%25A8%25B0" title="ਸ਼੍ਰੀਨਿਵਾਸ ਰਾਮਾਨੁਜਨ ਆਇੰਗਰ – Punjabi" lang="pa" hreflang="pa" data-title="ਸ਼੍ਰੀਨਿਵਾਸ ਰਾਮਾਨੁਜਨ ਆਇੰਗਰ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target">ਪੰਜਾਬੀ</a></li> <li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://pnb.wikipedia.org/wiki/%25D8%25B3%25D8%25B1%25DB%258C%25D9%2586%25D9%2588%25D8%25A7%25D8%25B3_%25D8%25B1%25D8%25A7%25D9%2585%25D8%25A7%25D9%2586%25D9%2588%25D8%25AC%25D9%2586" title="سرینواس رامانوجن – Western Punjabi" lang="pnb" hreflang="pnb" data-title="سرینواس رامانوجن" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target">پنجابی</a></li> <li class="interlanguage-link interwiki-ps mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ps.wikipedia.org/wiki/%25D8%25B3%25DB%258C%25D8%25B1%25DB%258C%25D9%2586%25DB%258C%25D9%2588%25D8%25A7%25D8%25B3%25D8%25A7_%25D8%25B1%25D8%25A7%25D9%2585%25D8%25A7%25D9%2586%25D9%2588%25D8%25AC%25D9%2586" title="سیرینیواسا رامانوجن – Pashto" lang="ps" hreflang="ps" data-title="سیرینیواسا رامانوجن" data-language-autonym="پښتو" data-language-local-name="Pashto" class="interlanguage-link-target">پښتو</a></li> <li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://pl.wikipedia.org/wiki/Srinivasa_Ramanujan" title="Srinivasa Ramanujan – Polish" lang="pl" hreflang="pl" data-title="Srinivasa Ramanujan" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li> <li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://pt.wikipedia.org/wiki/Sriniv%25C4%2581sa_R%25C4%2581m%25C4%2581nujan" title="Srinivāsa Rāmānujan – Portuguese" lang="pt" hreflang="pt" data-title="Srinivāsa Rāmānujan" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li> <li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ro.wikipedia.org/wiki/Srinivasa_Ramanujan" title="Srinivasa Ramanujan – Romanian" lang="ro" hreflang="ro" data-title="Srinivasa Ramanujan" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target">Română</a></li> <li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ru.wikipedia.org/wiki/%25D0%25A1%25D1%2580%25D0%25B8%25D0%25BD%25D0%25B8%25D0%25B2%25D0%25B0%25D1%2581%25D0%25B0_%25D0%25A0%25D0%25B0%25D0%25BC%25D0%25B0%25D0%25BD%25D1%2583%25D0%25B4%25D0%25B6%25D0%25B0%25D0%25BD" title="Сриниваса Рамануджан – Russian" lang="ru" hreflang="ru" data-title="Сриниваса Рамануджан" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li> <li class="interlanguage-link interwiki-sa mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://sa.wikipedia.org/wiki/%25E0%25A4%25B6%25E0%25A5%258D%25E0%25A4%25B0%25E0%25A5%2580%25E0%25A4%25A8%25E0%25A4%25BF%25E0%25A4%25B5%25E0%25A4%25BE%25E0%25A4%25B8%25E0%25A4%25B0%25E0%25A4%25BE%25E0%25A4%25AE%25E0%25A4%25BE%25E0%25A4%25A8%25E0%25A5%2581%25E0%25A4%259C%25E0%25A4%25A8%25E0%25A5%258D" title="श्रीनिवासरामानुजन् – Sanskrit" lang="sa" hreflang="sa" data-title="श्रीनिवासरामानुजन्" data-language-autonym="संस्कृतम्" data-language-local-name="Sanskrit" class="interlanguage-link-target">संस्कृतम्</a></li> <li class="interlanguage-link interwiki-skr mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://skr.wikipedia.org/wiki/%25D8%25B3%25D8%25B1%25DB%258C_%25D9%2586%25DB%258C%25D9%2588%25D8%25A7%25D8%25B3%25D8%25A7_%25D8%25B1%25D8%25A7%25D9%2585%25D8%25A7%25D9%2586%25D9%2588%25D8%25AC%25D9%2586" title="سری نیواسا رامانوجن – Saraiki" lang="skr" hreflang="skr" data-title="سری نیواسا رامانوجن" data-language-autonym="سرائیکی" data-language-local-name="Saraiki" class="interlanguage-link-target">سرائیکی</a></li> <li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://sq.wikipedia.org/wiki/Srinivasa_Ramanuxhan" title="Srinivasa Ramanuxhan – Albanian" lang="sq" hreflang="sq" data-title="Srinivasa Ramanuxhan" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target">Shqip</a></li> <li class="interlanguage-link interwiki-scn badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://scn.wikipedia.org/wiki/Srinivasa_Ramanujan" title="Srinivasa Ramanujan – Sicilian" lang="scn" hreflang="scn" data-title="Srinivasa Ramanujan" data-language-autonym="Sicilianu" data-language-local-name="Sicilian" class="interlanguage-link-target">Sicilianu</a></li> <li class="interlanguage-link interwiki-si mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://si.wikipedia.org/wiki/%25E0%25B7%2581%25E0%25B7%258A%25E2%2580%258D%25E0%25B6%25BB%25E0%25B7%2592%25E0%25B6%25B1%25E0%25B7%2592%25E0%25B7%2580%25E0%25B7%258F%25E0%25B7%2583%25E0%25B7%258A_%25E0%25B6%25BB%25E0%25B7%258F%25E0%25B6%25B8%25E0%25B7%258F%25E0%25B6%25B1%25E0%25B7%2594%25E0%25B6%25A2%25E0%25B6%25B1%25E0%25B7%258A" title="ශ්‍රිනිවාස් රාමානුජන් – Sinhala" lang="si" hreflang="si" data-title="ශ්‍රිනිවාස් රාමානුජන්" data-language-autonym="සිංහල" data-language-local-name="Sinhala" class="interlanguage-link-target">සිංහල</a></li> <li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://simple.wikipedia.org/wiki/Srinivasa_Ramanujan" title="Srinivasa Ramanujan – Simple English" lang="en-simple" hreflang="en-simple" data-title="Srinivasa Ramanujan" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target">Simple English</a></li> <li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://sk.wikipedia.org/wiki/Sr%25C3%25ADniv%25C3%25A1sa_Ajjang%25C3%25A1r_R%25C3%25A1m%25C3%25A1nud%25C5%25BEan" title="Srínivása Ajjangár Rámánudžan – Slovak" lang="sk" hreflang="sk" data-title="Srínivása Ajjangár Rámánudžan" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target">Slovenčina</a></li> <li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://sl.wikipedia.org/wiki/Srinivasa_Ajangar_Ramanud%25C5%25BEan" title="Srinivasa Ajangar Ramanudžan – Slovenian" lang="sl" hreflang="sl" data-title="Srinivasa Ajangar Ramanudžan" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target">Slovenščina</a></li> <li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://sr.wikipedia.org/wiki/%25D0%25A1%25D1%2580%25D0%25B8%25D0%25BD%25D0%25B8%25D0%25B2%25D0%25B0%25D1%2581%25D0%25B0_%25D0%25A0%25D0%25B0%25D0%25BC%25D0%25B0%25D0%25BD%25D1%2583%25D1%259F%25D0%25B0%25D0%25BD" title="Сриниваса Рамануџан – Serbian" lang="sr" hreflang="sr" data-title="Сриниваса Рамануџан" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target">Српски / srpski</a></li> <li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://sh.wikipedia.org/wiki/Srinivasa_Ramanud%25C5%25BEan" title="Srinivasa Ramanudžan – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Srinivasa Ramanudžan" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target">Srpskohrvatski / српскохрватски</a></li> <li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://fi.wikipedia.org/wiki/Srinivasa_Ramanujan" title="Srinivasa Ramanujan – Finnish" lang="fi" hreflang="fi" data-title="Srinivasa Ramanujan" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target">Suomi</a></li> <li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://sv.wikipedia.org/wiki/Srinivasa_Aiyangar_Ramanujan" title="Srinivasa Aiyangar Ramanujan – Swedish" lang="sv" hreflang="sv" data-title="Srinivasa Aiyangar Ramanujan" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target">Svenska</a></li> <li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://tl.wikipedia.org/wiki/Srinivasa_Ramanujan" title="Srinivasa Ramanujan – Tagalog" lang="tl" hreflang="tl" data-title="Srinivasa Ramanujan" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target">Tagalog</a></li> <li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ta.wikipedia.org/wiki/%25E0%25AE%259A%25E0%25AF%2580%25E0%25AE%25A9%25E0%25AE%25BF%25E0%25AE%25B5%25E0%25AE%25BE%25E0%25AE%259A_%25E0%25AE%2587%25E0%25AE%25B0%25E0%25AE%25BE%25E0%25AE%25AE%25E0%25AE%25BE%25E0%25AE%25A9%25E0%25AF%2581%25E0%25AE%259A%25E0%25AE%25A9%25E0%25AF%258D" title="சீனிவாச இராமானுசன் – Tamil" lang="ta" hreflang="ta" data-title="சீனிவாச இராமானுசன்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target">தமிழ்</a></li> <li class="interlanguage-link interwiki-te mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://te.wikipedia.org/wiki/%25E0%25B0%25B6%25E0%25B1%258D%25E0%25B0%25B0%25E0%25B1%2580%25E0%25B0%25A8%25E0%25B0%25BF%25E0%25B0%25B5%25E0%25B0%25BE%25E0%25B0%25B8_%25E0%25B0%25B0%25E0%25B0%25BE%25E0%25B0%25AE%25E0%25B0%25BE%25E0%25B0%25A8%25E0%25B1%2581%25E0%25B0%259C%25E0%25B0%25A8%25E0%25B1%258D" title="శ్రీనివాస రామానుజన్ – Telugu" lang="te" hreflang="te" data-title="శ్రీనివాస రామానుజన్" data-language-autonym="తెలుగు" data-language-local-name="Telugu" class="interlanguage-link-target">తెలుగు</a></li> <li class="interlanguage-link interwiki-th mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://th.wikipedia.org/wiki/%25E0%25B8%25A8%25E0%25B8%25A3%25E0%25B8%25B5%25E0%25B8%2599%25E0%25B8%25B4%25E0%25B8%25A7%25E0%25B8%25B2%25E0%25B8%25AA%25E0%25B8%25B0_%25E0%25B8%25A3%25E0%25B8%25B2%25E0%25B8%25A1%25E0%25B8%25B2%25E0%25B8%2599%25E0%25B8%25B8%25E0%25B8%258A%25E0%25B8%25B1%25E0%25B8%2599" title="ศรีนิวาสะ รามานุชัน – Thai" lang="th" hreflang="th" data-title="ศรีนิวาสะ รามานุชัน" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target">ไทย</a></li> <li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://tr.wikipedia.org/wiki/Srinivasa_Aiyangar_Ramanujan" title="Srinivasa Aiyangar Ramanujan – Turkish" lang="tr" hreflang="tr" data-title="Srinivasa Aiyangar Ramanujan" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target">Türkçe</a></li> <li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://uk.wikipedia.org/wiki/%25D0%25A1%25D1%2580%25D1%2596%25D0%25BD%25D1%2596%25D0%25B2%25D0%25B0%25D1%2581%25D0%25B0_%25D0%25A0%25D0%25B0%25D0%25BC%25D0%25B0%25D0%25BD%25D1%2583%25D0%25B4%25D0%25B6%25D0%25B0%25D0%25BD" title="Срініваса Рамануджан – Ukrainian" lang="uk" hreflang="uk" data-title="Срініваса Рамануджан" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target">Українська</a></li> <li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ur.wikipedia.org/wiki/%25D8%25B1%25D8%25A7%25D9%2585%25D8%25A7%25D9%2586%25D9%2588%25D8%25AC%25D9%2586" title="رامانوجن – Urdu" lang="ur" hreflang="ur" data-title="رامانوجن" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target">اردو</a></li> <li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://vi.wikipedia.org/wiki/Srinivasa_Ramanujan" title="Srinivasa Ramanujan – Vietnamese" lang="vi" hreflang="vi" data-title="Srinivasa Ramanujan" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target">Tiếng Việt</a></li> <li class="interlanguage-link interwiki-war mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://war.wikipedia.org/wiki/Srinivasa_Ramanujan" title="Srinivasa Ramanujan – Waray" lang="war" hreflang="war" data-title="Srinivasa Ramanujan" data-language-autonym="Winaray" data-language-local-name="Waray" class="interlanguage-link-target">Winaray</a></li> <li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://wuu.wikipedia.org/wiki/%25E6%2596%25AF%25E9%2587%258C%25E5%25B0%25BC%25E7%2593%25A6%25E7%2591%259F%25C2%25B7%25E6%258B%2589%25E9%25A9%25AC%25E5%258A%25AA%25E9%2587%2591" title="斯里尼瓦瑟·拉马努金 – Wu" lang="wuu" hreflang="wuu" data-title="斯里尼瓦瑟·拉马努金" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target">吴语</a></li> <li class="interlanguage-link interwiki-yo mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://yo.wikipedia.org/wiki/Srinivasa_Ramanujan" title="Srinivasa Ramanujan – Yoruba" lang="yo" hreflang="yo" data-title="Srinivasa Ramanujan" data-language-autonym="Yorùbá" data-language-local-name="Yoruba" class="interlanguage-link-target">Yorùbá</a></li> <li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://zh-yue.wikipedia.org/wiki/%25E6%258B%2589%25E9%25A6%25AC%25E5%258A%25AA%25E9%2587%2591" title="拉馬努金 – Cantonese" lang="yue" hreflang="yue" data-title="拉馬努金" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target">粵語</a></li> <li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://zh.wikipedia.org/wiki/%25E6%2596%25AF%25E9%2587%258C%25E5%25B0%25BC%25E7%2593%25A6%25E7%2591%259F%25C2%25B7%25E6%258B%2589%25E9%25A9%25AC%25E5%258A%25AA%25E9%2587%2591" title="斯里尼瓦瑟·拉马努金 – Chinese" lang="zh" hreflang="zh" data-title="斯里尼瓦瑟·拉马努金" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target">中文</a></li> </ul> </section> </div> <div class="minerva-footer-logo"> <img src="/static/images/mobile/copyright/wikipedia-wordmark-en.svg" alt="Wikipedia" width="120" height="18" 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<script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-57488d5c7d-v7wgc","wgBackendResponseTime":242,"wgPageParseReport":{"limitreport":{"cputime":"1.861","walltime":"2.203","ppvisitednodes":{"value":15366,"limit":1000000},"postexpandincludesize":{"value":341943,"limit":2097152},"templateargumentsize":{"value":18214,"limit":2097152},"expansiondepth":{"value":16,"limit":100},"expensivefunctioncount":{"value":14,"limit":500},"unstrip-depth":{"value":1,"limit":20},"unstrip-size":{"value":518812,"limit":5000000},"entityaccesscount":{"value":1,"limit":400},"timingprofile":["100.00% 1894.067 1 -total"," 27.22% 515.507 2 Template:Reflist"," 13.52% 256.077 41 Template:Cite_journal"," 9.32% 176.579 1 Template:Infobox_scientist"," 8.70% 164.714 40 Template:Cite_book"," 7.24% 137.207 35 Template:Cite_web"," 6.63% 125.606 1 Template:Efn"," 5.49% 103.919 1 Template:IPA-ta"," 5.42% 102.727 1 Template:IPA"," 4.22% 79.950 2 Template:Sfnp"]},"scribunto":{"limitreport-timeusage":{"value":"1.147","limit":"10.000"},"limitreport-memusage":{"value":23138810,"limit":52428800},"limitreport-logs":"anchor_id_list = table#1 {\n [\"CITEREFAlladiElliottGranville1998\"] = 1,\n [\"CITEREFAlyangar1995\"] = 1,\n [\"CITEREFBaaquieWilleboordse2009\"] = 1,\n [\"CITEREFBarraclough,_Leo2013\"] = 1,\n [\"CITEREFBasu2014\"] = 1,\n [\"CITEREFBell,_E._T.1928\"] = 1,\n [\"CITEREFBerndt1985\"] = 1,\n [\"CITEREFBerndt1989\"] = 1,\n [\"CITEREFBerndt1993\"] = 1,\n [\"CITEREFBerndt1997\"] = 2,\n [\"CITEREFBerndt1998\"] = 1,\n [\"CITEREFBerndt1999\"] = 1,\n [\"CITEREFBerndt2004\"] = 1,\n [\"CITEREFBerndt2005\"] = 1,\n [\"CITEREFBerndt2006\"] = 1,\n [\"CITEREFBerndtRankin1995\"] = 1,\n [\"CITEREFBerndtRankin2000\"] = 1,\n [\"CITEREFBerndtRankin2001\"] = 1,\n [\"CITEREFBiswas2006\"] = 1,\n [\"CITEREFBressoud,_David2006\"] = 1,\n [\"CITEREFBruce_C._BerndtRobert_A._Rankin2000\"] = 1,\n [\"CITEREFBullough1990\"] = 1,\n [\"CITEREFCalyampudi_Radhakrishna_Rao1997\"] = 1,\n [\"CITEREFChaitin2007\"] = 1,\n [\"CITEREFClark\"] = 1,\n [\"CITEREFDATTA2014\"] = 1,\n [\"CITEREFEmbleton2018\"] = 1,\n [\"CITEREFFreudenberger,_Nell2007\"] = 1,\n [\"CITEREFGunn,_J._W._C.Savage,_B.1919\"] = 1,\n [\"CITEREFHardy\"] = 1,\n [\"CITEREFHardy1910\"] = 1,\n [\"CITEREFHardy1920\"] = 1,\n [\"CITEREFHardy1937\"] = 2,\n [\"CITEREFHardy1940\"] = 2,\n [\"CITEREFHardy1978\"] = 1,\n [\"CITEREFHardy1979\"] = 1,\n [\"CITEREFHardy1999\"] = 2,\n [\"CITEREFHardyRamanujan1917\"] = 1,\n [\"CITEREFHardyRamanujan1918\"] = 2,\n [\"CITEREFHarvey2003\"] = 1,\n [\"CITEREFHenderson1995\"] = 1,\n [\"CITEREFHerschfeld1935\"] = 1,\n [\"CITEREFIwaniec1989\"] = 1,\n [\"CITEREFJames_R._Newman2000\"] = 1,\n [\"CITEREFJanardhanan2015\"] = 1,\n [\"CITEREFKanigel1991\"] = 1,\n [\"CITEREFKanigel2004\"] = 1,\n [\"CITEREFKolata1987\"] = 1,\n [\"CITEREFKrishnamachari2013\"] = 2,\n [\"CITEREFKrishnamurthy\"] = 1,\n [\"CITEREFKumar2018\"] = 1,\n [\"CITEREFLangley1921\"] = 1,\n [\"CITEREFLeavitt2007\"] = 1,\n [\"CITEREFLunden2010\"] = 1,\n [\"CITEREFMcNary,_Dave2014\"] = 1,\n [\"CITEREFNarlikar2003\"] = 1,\n [\"CITEREFNeville1921\"] = 1,\n [\"CITEREFNeville1942\"] = 1,\n [\"CITEREFOlaussonSangster2006\"] = 1,\n [\"CITEREFOno2006\"] = 1,\n [\"CITEREFOnoAczel2016\"] = 2,\n [\"CITEREFPeterson\"] = 1,\n [\"CITEREFRam1972\"] = 1,\n [\"CITEREFRamanujan1914\"] = 2,\n [\"CITEREFRamanujan1915\"] = 4,\n [\"CITEREFRamanujan1916\"] = 3,\n [\"CITEREFRamanujan1917\"] = 1,\n [\"CITEREFRamanujan1919\"] = 2,\n [\"CITEREFRamanujan1920\"] = 1,\n [\"CITEREFRamanujan1921\"] = 1,\n [\"CITEREFRamanujan1968\"] = 1,\n [\"CITEREFRamanujanHardySeshu_AiyarWilson2000\"] = 1,\n [\"CITEREFRamanujan_AiyangarHardyAiyar2000\"] = 1,\n [\"CITEREFRanganathan1967\"] = 1,\n [\"CITEREFRao,_K._Srinivasa\"] = 1,\n [\"CITEREFRao,_K_Srinivasa\"] = 1,\n [\"CITEREFRibet,_Kenneth_A.2003\"] = 1,\n [\"CITEREFRoss2017\"] = 1,\n [\"CITEREFS._Ramanujan1957\"] = 1,\n [\"CITEREFS._Ramanujan1988\"] = 1,\n [\"CITEREFS._Ramanujan2012\"] = 1,\n [\"CITEREFSankaran2005\"] = 1,\n [\"CITEREFSeshu_Iyer1920\"] = 1,\n [\"CITEREFSnow1966\"] = 1,\n [\"CITEREFSrinivasan2012\"] = 1,\n [\"CITEREFTaylor,_D._J.2008\"] = 1,\n [\"CITEREFWinnie2019\"] = 1,\n [\"CITEREFWolfram2016\"] = 1,\n [\"CITEREFYoung,_D._A._B.1994\"] = 1,\n}\ntemplate_list = table#1 {\n [\"'\\\"\"] = 1,\n [\"Authority control\"] = 1,\n [\"Birth date\"] = 1,\n [\"Blockquote\"] = 4,\n [\"Cbignore\"] = 1,\n [\"Citation\"] = 1,\n [\"Cite ODNB\"] = 1,\n [\"Cite book\"] = 40,\n [\"Cite conference\"] = 1,\n [\"Cite journal\"] = 41,\n [\"Cite magazine\"] = 2,\n [\"Cite news\"] = 16,\n [\"Cite report\"] = 1,\n [\"Cite web\"] = 34,\n [\"Colend\"] = 1,\n [\"Convert\"] = 3,\n [\"DEFAULTSORT:Srinivasa, Ramanujan\"] = 1,\n [\"Death date and age\"] = 1,\n [\"Div col\"] = 2,\n [\"Div col end\"] = 1,\n [\"Efn\"] = 1,\n [\"Further\"] = 2,\n [\"Good article\"] = 1,\n [\"Harvnb\"] = 49,\n [\"IPA-ta\"] = 1,\n [\"IPAc-en\"] = 1,\n [\"ISBN\"] = 6,\n [\"Indian mathematics\"] = 1,\n [\"Indian name\"] = 1,\n [\"Infobox scientist\"] = 1,\n [\"Isbn\"] = 1,\n [\"MacTutor\"] = 1,\n [\"Main\"] = 2,\n [\"Math\"] = 18,\n [\"MathGenealogy\"] = 1,\n [\"Multiple image\"] = 1,\n [\"Mvar\"] = 10,\n [\"Notelist\"] = 1,\n [\"Nowrap\"] = 6,\n [\"OEIS\"] = 1,\n [\"Pad\"] = 1,\n [\"Pi\"] = 4,\n [\"Plainlist\"] = 3,\n [\"Portal\"] = 1,\n [\"Postnominals\"] = 2,\n [\"Pp-vandalism\"] = 1,\n [\"Quote box\"] = 2,\n [\"Redirect\"] = 1,\n [\"Refbegin\"] = 2,\n [\"Refend\"] = 2,\n [\"Reflist\"] = 1,\n [\"Respell\"] = 1,\n [\"Rp\"] = 2,\n [\"ScienceWorldBiography\"] = 1,\n [\"Sfnp\"] = 2,\n [\"Sfrac\"] = 4,\n [\"Short description\"] = 1,\n [\"Sister project links\"] = 1,\n [\"Spaced ndash\"] = 1,\n [\"Sqrt\"] = 1,\n [\"Use British English\"] = 1,\n [\"Use dmy dates\"] = 1,\n [\"Webarchive\"] = 4,\n [\"Xref\"] = 1,\n}\narticle_whitelist = table#1 {\n}\n","limitreport-profile":[["?","260","23.6"],["MediaWiki\\Extension\\Scribunto\\Engines\\LuaSandbox\\LuaSandboxCallback::callParserFunction","140","12.7"],["MediaWiki\\Extension\\Scribunto\\Engines\\LuaSandbox\\LuaSandboxCallback::gsub","140","12.7"],["dataWrapper \u003Cmw.lua:672\u003E","120","10.9"],["MediaWiki\\Extension\\Scribunto\\Engines\\LuaSandbox\\LuaSandboxCallback::match","60","5.5"],["MediaWiki\\Extension\\Scribunto\\Engines\\LuaSandbox\\LuaSandboxCallback::find","60","5.5"],["\u003Cmw.lua:694\u003E","40","3.6"],["(for generator)","20","1.8"],["MediaWiki\\Extension\\Scribunto\\Engines\\LuaSandbox\\LuaSandboxCallback::preprocess","20","1.8"],["MediaWiki\\Extension\\Scribunto\\Engines\\LuaSandbox\\LuaSandboxCallback::getAllExpandedArguments","20","1.8"],["[others]","220","20.0"]]},"cachereport":{"origin":"mw-web.codfw.main-57488d5c7d-wfv5x","timestamp":"20241128015344","ttl":2592000,"transientcontent":false}}});});</script> <script 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Srinivasa Ramanujan - Wikipedia