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{{Short description|British mathematician (1885–1977)}}
{{Use British English|date=August 2014}}
{{Use dmy dates|date=August 2014}}
{{Infobox scientist
| name              = John E. Littlewood
| honorific_suffix  = {{post-nominals|country=GBR|FRS|size=100%}}
| image             = John Edensor Littlewood.jpg
| caption           = 
| birth_name        = John Edensor Littlewood
| birth_date        = {{birth date|df=yes|1885|6|9}}
| birth_place       = [[Rochester, Kent|Rochester]], [[Kent]], England
| death_date        = {{death date and age|df=yes|1977|9|6|1885|6|9}}
| death_place       = [[Cambridge]], England
| field             = Mathematics
| work_institution  = [[University of Cambridge]]
| alma_mater        = [[Trinity College, Cambridge]]
| doctoral_advisor  = [[Ernest Barnes]]
| doctoral_students = {{hlist | [[A.&nbsp;O.&nbsp;L. Atkin]] | [[Sarvadaman Chowla]] | [[Harold Davenport]] | [[Srinivasa Ramanujan]] | [[Stanley Skewes]] | [[Donald C. Spencer]] | [[Albert Ingham]]}}
| known_for         = [[Mathematical analysis]]
| prizes            = {{ubl | [[Smith's Prize]] (1908) | [[Royal Medal]] (1929) | [[De Morgan Medal]] (1938) | [[Sylvester Medal]] (1943) | [[Copley Medal]] (1958) | [[Senior Berwick Prize]] (1960)}}
}}
'''John Edensor Littlewood''' {{post-nominals|country=GBR|FRS}} (9 June 1885 – 6 September 1977) was a British mathematician. He worked on topics relating to [[mathematical analysis|analysis]], [[number theory]], and [[differential equation]]s and had lengthy collaborations with [[G.&nbsp;H. Hardy]], [[Srinivasa Ramanujan]] and [[Mary Cartwright]].

==Biography  ==
Littlewood was born on the 9th of June 1885 in [[Rochester, Kent]], the eldest son of Edward Thornton Littlewood and Sylvia Maud (née Ackland).{{Sfn|Burkill|1978|p=322}} In 1892, his father accepted the headmastership of a school in [[Wynberg, Cape Town]], in South Africa, taking his family there.<ref>{{Harvnb|Burkill|1978|p=324}}: "He later accepted the headmastership of a newly founded school at Wynberg near Cape Town, taking his family there in 1892."</ref> Littlewood returned to Britain in 1900 to attend [[St Paul's School, London|St Paul's School]] in London, studying under [[Francis Sowerby Macaulay]], an influential [[algebraic geometry|algebraic geometer]].<ref>{{Harvnb|Bateman|Diamond|1978|p=28}}: "In 1900 he returned to England, where he attended [[St Paul's School, London|St. Paul's School]] and studied with the talented teacher and mathematician [[Francis Sowerby Macaulay|F. S. Macaulay]]."</ref>

In 1903, Littlewood entered the [[University of Cambridge]], studying in [[Trinity College, Cambridge|Trinity College]]. He spent his first two years preparing for the [[Tripos]] examinations which qualify undergraduates for a bachelor's degree where he emerged in 1905 as [[Senior Wrangler (University of Cambridge)|Senior Wrangler]] bracketed with [[James Mercer (mathematician)|James Mercer]] (Mercer had already graduated from the [[University of Manchester]] before attending Cambridge<ref>{{Harvnb|Burkill|1978|p=325}}</ref>). In 1906, after completing the second part of the Tripos, he started his research under [[Ernest Barnes]].<ref>{{Harvnb|Bateman|Diamond|1978|pp=28–29}}: "He began his research later that year on asymptotic formulas for integral functions of order zero, under his tutor and director of studies [[Ernest Barnes|E. W. Barnes]]."</ref> One of the problems that Barnes suggested to Littlewood was to prove the [[Riemann hypothesis]], an assignment at which he did not succeed.<ref>{{Harvnb|Bateman|Diamond|1978|p=29}}: "Barnes proposed to Littlewood the task of proving the Riemann hypothesis ... he did not succeed in that strenuous assignment ..."</ref> He was elected a Fellow of Trinity College in 1908. From October 1907 to June 1910, he worked as a [[Richardson Chair of Applied Mathematics|Richardson Lecturer]] in the [[School of Mathematics, University of Manchester|School of Mathematics]] at the University of Manchester before returning to Cambridge in October 1910, where he remained for the rest of his career. He was appointed [[Rouse Ball Professor of Mathematics]] in 1928, retiring in 1950.  He was elected a [[Fellow of the Royal Society]] in 1916, awarded the [[Royal Medal]] in 1929, the [[Sylvester Medal]] in 1943, and the [[Copley Medal]] in 1958.  He was president of the [[London Mathematical Society]] from 1941 to 1943 and was awarded the [[De Morgan Medal]] in 1938 and the [[Berwick Prizes|Senior Berwick Prize]] in 1960.

Littlewood died on 6 September 1977.

==Work==
Most of Littlewood's work was in the field of [[mathematical analysis]]. He began research under the supervision of [[Ernest William Barnes]], who suggested that he attempt to prove the [[Riemann hypothesis]]: Littlewood showed that if the Riemann hypothesis is true, then the [[prime number theorem]] follows and obtained the error term.  This work won him his Trinity fellowship. However, the link between the Riemann hypothesis and the prime number theorem had been known before in Continental Europe, and Littlewood wrote later in his book, ''A Mathematician's Miscellany'' that his rediscovery of the result did not shed a positive light on the isolated nature of British mathematics at the time.<ref name="Littlewood Miscellany Bollobás ">{{cite book | last=Littlewood | first=John Edensor |  editor-last=Bollobás | editor-first=Béla | title=Littlewood's Miscellany | publisher=Cambridge University Press | publication-place=Cambridge New York Port Chester [etc.] | date=1986-10-30 | isbn=0-521-33702-X | page=89}}</ref><ref>{{Harvnb|Bateman|Diamond|1978|p=29}}: "This heroic proposal and Littlewood's later account of it are commentaries upon the isolation of British mathematics at that time."</ref>

===Theory of the distribution of prime numbers===
In 1914, Littlewood published his first result in the field of [[analytic number theory]] concerning the error term of the [[prime-counting function]]. If {{math|''π''(''x'')}} denotes the number of primes up {{math|''x''}}, then the [[prime number theorem]] implies that {{math|''π''(''x'') ~ Li(''x'')}},
where <math> \operatorname{Li}(x) = \int_2^x \frac{\mathrm{d}t}{\log t}</math> is known as the [[logarithmic integral#offset logarithmic integral|Eulerian logarithmic integral]].
Numerical evidence seemed to suggest that {{math|{{pi}}(''x'')&nbsp;<&nbsp;Li(''x'')}} for all {{math|''x''}}. Littlewood, however proved<ref>{{cite journal |first=J. E.|last= Littlewood | author-link=J. E. Littlewood | title=Sur la distribution des nombres premiers|journal=[[Comptes Rendus]]|volume= 158 |year=1914|pages= 1869–1872 | jfm=45.0305.01}}</ref> that the difference {{math|{{pi}}(''x'')&nbsp;−&nbsp;Li(''x'')}} changes sign infinitely often.

===Collaboration with G. H. Hardy===
Littlewood collaborated for many years with [[G. H. Hardy]]. Together they devised the [[first Hardy–Littlewood conjecture]], a strong form of the [[twin prime conjecture]], and the [[second Hardy–Littlewood conjecture]].

====Ramanujan====
He also, with Hardy, identified the work of the Indian mathematician [[Srinivasa Ramanujan]] as that of a genius and supported him in travelling from India to work at Cambridge.<ref>Hardy (June 1920), pp&nbsp;494–495.</ref> A self-taught mathematician, Ramanujan later became a [[Fellow of the Royal Society]], Fellow of [[Trinity College, Cambridge]], and widely recognised as on a par with other geniuses such as [[Euler]] and [[Carl Gustav Jacob Jacobi|Jacobi]].<ref>{{Cite book|last=Hardy |first=G. H. |title= Collected Papers of G. H. Hardy|year= 1979 | isbn=978-0-19-853347-4 |publisher= [[Oxford University Press|Clarendon Press]] |location=Oxford, England| page=720|volume=7|no-pp=true}}</ref>

===Collaboration with Mary Cartwright===

In the late 1930s, as the prospect of war loomed, the [[Department of Scientific and Industrial Research (United Kingdom)|Department of Scientific and Industrial Research]] sought the interest of pure mathematicians in the properties of [[differential equations|non linear differential equations]] that were needed by radio engineers and scientists. The problems appealed to Littlewood and [[Mary Cartwright]], and they worked on them independently during the next 20 years.<ref>{{Harvnb|Burkill|1978|p=322}}</ref>

The problems that Littlewood and Cartwright worked on concerned [[differential equations]] arising out of early research on [[radar]]: their work foreshadowed the modern theory of dynamical systems.  [[Littlewood's 4/3 inequality]] on bilinear forms was a forerunner of the later [[Grothendieck]] [[tensor norm]] theory.

==Military service WWI – ballistics work==
During the [[First World War|Great War]], Littlewood served in the [[Royal Garrison Artillery]] as a second lieutenant. He made highly significant contributions in the field of ballistics.<ref>J. E. Littlewood, Adventures in ballistics, 1915–1918, I, Math. Spectrum 4
(1971/72)</ref><ref>J. E. Littlewood, Adventures in ballistics, 1915–1918, II, Math. Spectrum
4 (1971/72), 80–86.</ref>

==Later life==

He continued to write papers into his eighties, particularly in analytical areas of what would become the theory of [[dynamical systems]].

Littlewood is also remembered for his book of reminiscences, ''[[A Mathematician's Miscellany]]'' (new edition published in 1986).

Among his PhD students were [[Sarvadaman Chowla]], [[Harold Davenport]], and [[Donald C. Spencer]]. Spencer reported that in 1941 when he (Spencer) was about to get on the boat that would take him home to the United States, Littlewood reminded him: "''n'', ''n'' alpha, ''n'' beta!" (referring to [[Littlewood's conjecture]]).

Littlewood's collaborative work, carried out by correspondence, covered fields in [[Diophantine approximation]] and [[Waring's problem]], in particular. In his other work, he collaborated with [[Raymond Paley]] on [[Littlewood–Paley theory]] in [[Fourier transform|Fourier theory]], and with [[Cyril Offord]] in combinatorial work on random sums, in developments that opened up fields that are still intensively studied.

In a 1947 lecture, the Danish mathematician [[Harald Bohr]] said, "To illustrate to what extent Hardy and Littlewood in the course of the years came to be considered as the leaders of recent English mathematical research, I may report what an excellent colleague once jokingly said: 'Nowadays, there are only three really great English mathematicians: Hardy, Littlewood, and Hardy–Littlewood.'{{thin space}}"<ref>
{{cite book
|last=Bohr
|first=Harald
|author-link=Harald Bohr
|title=Collected Mathematical Works
|volume=1
|year=1952
|publisher=Dansk Matematisk Forening
|location=Copenhagen
|oclc=3172542
|pages=xiii–xxxiv
|chapter=Looking Backward
|no-pp=true
}}
</ref> {{Rp|xxvii}}

The German mathematician [[Edmund Landau]] supposed that Littlewood was a pseudonym that Hardy used for his lesser work and "so doubted the existence of Littlewood that he made a special trip to Great Britain to see the man with his own eyes".<ref name="Krantz 1990 pp. 32–38">{{cite journal | last=Krantz | first=Steven G. | title=Mathematical anecdotes | journal=The Mathematical Intelligencer | volume=12 | issue=4 | date=1990 | issn=0343-6993 | doi=10.1007/BF03024029 | pages=32–38}}</ref>  He visited Cambridge where he saw much of Hardy but nothing of Littlewood and so considered his conjecture to be proven.  A similar story was told about [[Norbert Wiener]], who vehemently denied it in his autobiography.<ref>{{citation |journal=The American Mathematical Monthly |volume=96 |year=1989 |number=2 |title=Littlewood's Miscellany |author=Ralph P. Boas |pages=167–169 |doi=10.1080/00029890.1989.11972165}}</ref>  

He coined [[Littlewood's law]], which states that individuals can expect "miracles" to happen to them at the rate of about one per month.

==Cultural references==
John Littlewood is depicted in two films covering the life of Ramanujan – ''[[Ramanujan (film)|Ramanujan]]'' in 2014 portrayed by [[Michael Lieber]] and ''[[The Man Who Knew Infinity (film)|The Man Who Knew Infinity]]'' in 2015 portrayed by [[Toby Jones]].

==See also==
* [[Critical line theorem]]
* [[Littlewood conjecture]]
* [[Littlewood polynomial]]
* [[Littlewood's three principles of real analysis]]
* [[Littlewood's Tauberian theorem]]
* [[Littlewood's 4/3 inequality]]
* [[Littlewood subordination theorem]]
* [[Littlewood–Offord problem]]
* [[Littlewood–Paley theory]]
* [[Hardy–Littlewood circle method]]
* [[Big O notation#The Hardy–Littlewood definition|Hardy–Littlewood definition]]
* [[Hardy–Littlewood inequality]]
* [[Hardy–Littlewood maximal function]]
* [[Hardy–Littlewood zeta function conjectures]]
* [[Hardy–Littlewood tauberian theorem]]
* [[Twin prime#First Hardy–Littlewood conjecture|First Hardy–Littlewood conjecture]]
* [[Second Hardy–Littlewood conjecture]]
* [[Ross–Littlewood paradox]]
* [[Hadamard three-circle theorem]]
* [[Skewes's number]]

==References==
{{reflist}}

===Bibliography===
* {{Cite journal | last1 = Burkill | first1 = J. C. | doi = 10.1098/rsbm.1978.0010 | title = John Edensor Littlewood. 9 June 1885–6 September 1977 | journal = Biographical Memoirs of Fellows of the Royal Society | volume = 24 | pages = 322–326 | year = 1978 | jstor = 769763| s2cid = 119754496 }}
* {{cite journal | title=John E. Littlewood (1885–1977) An Informal Obituary | last1=Bateman | first1=P | last2=Diamond | first2=P | journal=[[The Mathematical Intelligencer]] | year=1978 | volume=1 | issue=1 | pages=28–33 | doi=10.1007/BF03023041| s2cid=122107252 }}

==Further reading==
* ''Littlewood's Miscellany'', edited by [[B. Bollobás]], [[Cambridge University Press]]; 1986. {{isbn|0-521-33702-X}} (alternative title for [[A Mathematician's Miscellany]])

==External links==
{{Wikiquote|John Littlewood|John Edensor Littlewood}}
* {{MacTutor Biography|id=Littlewood}}
* {{MathGenealogy|id=10463}}
* [http://www.numbertheory.org/obituaries/LMS/littlewood/index.html Papers of Littlewood on Number Theory]
* [https://archive.org/details/mathematiciansmi033496mbp A Mathematicians Miscellany]

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{{Copley Medallists 1951–2000}}
{{De Morgan Medallists}}

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{{DEFAULTSORT:Littlewood, John Edensor}}
[[Category:British number theorists]]
[[Category:Mathematical analysts]]
[[Category:20th-century English  mathematicians]]
[[Category:Recipients of the Copley Medal]]
[[Category:Fellows of Trinity College, Cambridge]]
[[Category:Fellows of the Royal Society]]
[[Category:Alumni of Trinity College, Cambridge]]
[[Category:People educated at St Paul's School, London]]
[[Category:People from Rochester, Kent]]
[[Category:1885 births]]
[[Category:1977 deaths]]
[[Category:Royal Medal winners]]
[[Category:Senior Wranglers]]
[[Category:De Morgan Medallists]]
[[Category:Royal Garrison Artillery officers]]
[[Category:Rouse Ball Professors of Mathematics (Cambridge)]]