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	==See also==		==See also==
	~~{{Portal\|Mathematics\|Biography\|India}}~~
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← Older revision		Revision as of 09:33, 22 July 2023
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	\| birth_name = Srinivasa Ramanujan Aiyangar		\| birth_name = Srinivasa Ramanujan Aiyangar
	\| birth_date = {{Birth date\|df=yes\|1887\|12\|22}}		\| birth_date = {{Birth date\|df=yes\|1887\|12\|22}}
	\| birth_place = [[Erode]], [[Mysore Princely State\|Mysore State]], [[British India]]		\| birth_place = [[Erode]], [[Mysore Princely State\|Mysore State]], [[British India]] (now in [[Tamil Nadu]], [[India]])
	\| death_date = {{Death date and age\|df=yes\|1920\|4\|26\|1887\|12\|22}}		\| death_date = {{Death date and age\|df=yes\|1920\|4\|26\|1887\|12\|22}}
	\| death_place = [[Kumbakonam]], [[Madras Presidency\|Madras]], British India		\| death_place = [[Kumbakonam]], [[Tanjore District (Madras Presidency)\|Tanjore District]], [[Madras Presidency]], [[British India]] (now [[Thanjavur district]],<br/>[[Tamil Nadu]], [[India]])
	\| citizenship = [[British Raj\|British ~~India~~]]		\| citizenship = [[British Raj\|British Indian]]
	\| workplaces = [[Trinity College, Cambridge]]		\| workplaces = [[Trinity College, Cambridge]]
	\| education = {{plainlist\|		\| education = {{plainlist\|
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	{{Main\|Ramanujan–Petersson conjecture}}		{{Main\|Ramanujan–Petersson conjecture}}

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

	In his paper "On certain arithmetical functions", Ramanujan defined the so-called delta-function, whose coefficients are called {{math\|''τ''(''n'')}} (the [[Ramanujan tau function]]).<ref>{{cite journal \|last= Ramanujan \|first= Srinivasa \|title= On certain arithmetical functions \|journal= Transactions of the Cambridge Philosophical Society \|volume=XXII \|number= 9 \|year= 1916 \|url= https://ramanujan.sirinudi.org/Volumes/published/ram18.pdf \|access-date= 15 May 2016 \|url-status=dead \|archive-url= https://web.archive.org/web/20160611175341/https://ramanujan.sirinudi.org/Volumes/published/ram18.pdf \|archive-date= 11 June 2016 \|df= dmy-all }} The tau function is discussed in pages 194–197.</ref> He proved many congruences for these numbers, such as {{math\|''τ''(''p'') ≡ 1 + ''p''<sup>11</sup> mod 691}} for primes {{mvar\|p}}. This congruence (and others like it that Ramanujan proved) inspired [[Jean-Pierre Serre]] (1954 Fields Medalist) to conjecture that there is a theory of [[Galois module\|Galois representations]] that "explains" these congruences and more generally all modular forms. {{math\|Δ(''z'')}} is the first example of a modular form to be studied in this way. Deligne (in his Fields Medal-winning work) proved Serre's conjecture. The proof of [[Fermat's Last Theorem]] proceeds by first reinterpreting [[elliptic curve]]s and modular forms in terms of these Galois representations. Without this theory, there would be no proof of Fermat's Last Theorem.<ref>{{cite book\|last1= Ono \|first1= Ken \|author-link1= Ken Ono \|last2= Aczel\|first2= Amir D. \|author-link2= Amir Aczel \|title= My Search for Ramanujan: How I Learned to Count \|date= 13 April 2016 \|publisher= [[Springer Science+Business Media\|Springer]] \|pages= 236–237 \|isbn= 978-3319255668 \|url= https://books.google.com/books?id=JZkFDAAAQBAJ&q=%22ideas+that+were+critical+to+the+proof+of+Fermat%27s+last+theorem%22&pg=PA236 \|quote= ideas that were critical to the proof of Fermat's last theorem}}</ref>		In his paper "On certain arithmetical functions", Ramanujan defined the so-called delta-function, whose coefficients are called {{math\|''τ''(''n'')}} (the [[Ramanujan tau function]]).<ref>{{cite journal \|last= Ramanujan \|first= Srinivasa \|title= On certain arithmetical functions \|journal= Transactions of the Cambridge Philosophical Society \|volume=XXII \|number= 9 \|year= 1916 \|url= https://ramanujan.sirinudi.org/Volumes/published/ram18.pdf \|access-date= 15 May 2016 \|url-status=dead \|archive-url= https://web.archive.org/web/20160611175341/https://ramanujan.sirinudi.org/Volumes/published/ram18.pdf \|archive-date= 11 June 2016 \|df= dmy-all }} The tau function is discussed in pages 194–197.</ref> He proved many congruences for these numbers, such as {{math\|''τ''(''p'') ≡ 1 + ''p''<sup>11</sup> mod 691}} for primes {{mvar\|p}}. This congruence (and others like it that Ramanujan proved) inspired [[Jean-Pierre Serre]] (1954 Fields Medalist) to conjecture that there is a theory of [[Galois module\|Galois representations]] that "explains" these congruences and more generally all modular forms. {{math\|Δ(''z'')}} is the first example of a modular form to be studied in this way. Deligne (in his Fields Medal-winning work) proved Serre's conjecture. The proof of [[Fermat's Last Theorem]] proceeds by first reinterpreting [[elliptic curve]]s and modular forms in terms of these Galois representations. Without this theory, there would be no proof of Fermat's Last Theorem.<ref>{{cite book\|last1= Ono \|first1= Ken \|author-link1= Ken Ono \|last2= Aczel\|first2= Amir D. \|author-link2= Amir Aczel \|title= My Search for Ramanujan: How I Learned to Count \|date= 13 April 2016 \|publisher= [[Springer Science+Business Media\|Springer]] \|pages= 236–237 \|isbn= 978-3319255668 \|url= https://books.google.com/books?id=JZkFDAAAQBAJ&q=%22ideas+that+were+critical+to+the+proof+of+Fermat%27s+last+theorem%22&pg=PA236 \|quote= ideas that were critical to the proof of Fermat's last theorem}}</ref>
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	{{blockquote\|He combined a power of generalisation, a feeling for form, and a capacity for rapid modification of his hypotheses, that were often really startling, and made him, in his own peculiar field, without a rival in his day. The limitations of his knowledge were as startling as its profundity. Here was a man who could work out [[modular equation]]s and theorems... to orders unheard of, whose mastery of continued fractions was... beyond that of any mathematician in the world, who had found for himself the functional equation of the [[Riemann zeta function\|zeta function]] and the dominant terms of many of the most famous problems in the analytic theory of numbers; and yet he had never heard of a [[doubly periodic function]] or of [[Cauchy's integral theorem\|Cauchy's theorem]], and had indeed but the vaguest idea of what a function of a [[complex variable]] was..."}}		{{blockquote\|He combined a power of generalisation, a feeling for form, and a capacity for rapid modification of his hypotheses, that were often really startling, and made him, in his own peculiar field, without a rival in his day. The limitations of his knowledge were as startling as its profundity. Here was a man who could work out [[modular equation]]s and theorems... to orders unheard of, whose mastery of continued fractions was... beyond that of any mathematician in the world, who had found for himself the functional equation of the [[Riemann zeta function\|zeta function]] and the dominant terms of many of the most famous problems in the analytic theory of numbers; and yet he had never heard of a [[doubly periodic function]] or of [[Cauchy's integral theorem\|Cauchy's theorem]], and had indeed but the vaguest idea of what a function of a [[complex variable]] was..."}}

	When asked about the methods Ramanujan employed to arrive at his solutions, Hardy said they were "arrived at by a process of mingled argument, intuition, and induction, of which he was entirely unable to give any coherent account."<ref name="cola1">[https://www.usna.edu/Users/math/meh/ramanujan.html Srinivasa Ramanujan] {{webarchive\|url=https://web.archive.org/web/20050325041218/https://www.usna.edu/Users/math/meh/ramanujan.html \|date=25 March 2005 }}. Retrieved 2 December 2010.</ref> He also said that he had "never met his equal, and can compare him only with Euler or Jacobi".<ref name="cola1"/> Littlewood reportedly said that helping Ramanujan catch up to European mathematics beyond what was available in India was very difficult, because each new point mentioned to Ramanujan caused him to produce original ideas that prevented Littlewood from continuing the lesson.<ref name="alyangar1995">{{Cite book \|last=Alyangar \|first=Srinivasa Ramanujan \|url=https://~~www~~.google.com/books~~/edition/Ramanujan/Of5G0r6DQiEC~~?hl=~~en&gbpv=1~~&pg=PA51~~&printsec=frontcover~~ \|title=Ramanujan: Letters and Commentary \|publisher=American Mathematical Society \|year=1995 \|isbn=0821891251 \|editor-last=Berndt \|editor-first=Bruce C. \|series=History of Mathematics \|volume=9 \|page=51 \|access-date=2023-02-22 \|editor-last2=Rankin \|editor-first2=Robert Alexander}}</ref>		When asked about the methods Ramanujan employed to arrive at his solutions, Hardy said they were "arrived at by a process of mingled argument, intuition, and induction, of which he was entirely unable to give any coherent account."<ref name="cola1">[https://www.usna.edu/Users/math/meh/ramanujan.html Srinivasa Ramanujan] {{webarchive\|url=https://web.archive.org/web/20050325041218/https://www.usna.edu/Users/math/meh/ramanujan.html \|date=25 March 2005 }}. Retrieved 2 December 2010.</ref> He also said that he had "never met his equal, and can compare him only with Euler or Jacobi".<ref name="cola1"/> Littlewood reportedly said that helping Ramanujan catch up with European mathematics beyond what was available in India was very difficult, because each new point mentioned to Ramanujan caused him to produce original ideas that prevented Littlewood from continuing the lesson.<ref name="alyangar1995">{{Cite book \|last=Alyangar \|first=Srinivasa Ramanujan \|url=https://books.google.com/books?id=Of5G0r6DQiEC&pg=PA51 \|title=Ramanujan: Letters and Commentary \|publisher=American Mathematical Society \|year=1995 \|isbn=0821891251 \|editor-last=Berndt \|editor-first=Bruce C. \|series=History of Mathematics \|volume=9 \|page=51 \|access-date=2023-02-22 \|editor-last2=Rankin \|editor-first2=Robert Alexander}}</ref>

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

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