John Edensor LittlewoodFRS (9 June 1885 – 6 September 1977) was a British mathematician. He worked on topics relating toanalysis,number theory, anddifferential equations and had lengthy collaborations withG. H. Hardy,Srinivasa Ramanujan andMary Cartwright.

John E. Littlewood FRS
Born	John Edensor Littlewood (1885-06-09)9 June 1885 Rochester,Kent, England
Died	6 September 1977(1977-09-06) (aged 92) Cambridge, England
Alma mater	Trinity College, Cambridge
Known for	Mathematical analysis
Awards	Smith's Prize (1908) Royal Medal (1929) De Morgan Medal (1938) Sylvester Medal (1943) Copley Medal (1958) Senior Berwick Prize (1960)
Scientific career
Fields	Mathematics
Institutions	University of Cambridge
Doctoral advisor	Ernest Barnes
Doctoral students	A. O. L. Atkin Sarvadaman Chowla Harold Davenport Srinivasa Ramanujan Stanley Skewes Donald C. Spencer Albert Ingham

Littlewood was born on the 9th of June 1885 inRochester, Kent, the eldest son of Edward Thornton Littlewood and Sylvia Maud (née Ackland).^[1] In 1892, his father accepted the headmastership of a school inWynberg, Cape Town, in South Africa, taking his family there.^[2] Littlewood returned to Britain in 1900 to attendSt Paul's School in London, studying underFrancis Sowerby Macaulay, an influentialalgebraic geometer.^[3]

In 1903, Littlewood entered theUniversity of Cambridge, studying inTrinity College. He spent his first two years preparing for theTripos examinations which qualify undergraduates for a bachelor's degree where he emerged in 1905 asSenior Wrangler bracketed withJames Mercer (Mercer had already graduated from theUniversity of Manchester before attending Cambridge^[4]). In 1906, after completing the second part of the Tripos, he started his research underErnest Barnes.^[5] One of the problems that Barnes suggested to Littlewood was to prove theRiemann hypothesis, an assignment at which he did not succeed.^[6] He was elected a Fellow of Trinity College in 1908. From October 1907 to June 1910, he worked as aRichardson Lecturer in theSchool 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 appointedRouse Ball Professor of Mathematics in 1928, retiring in 1950. He was elected aFellow of the Royal Society in 1916, awarded theRoyal Medal in 1929, theSylvester Medal in 1943, and theCopley Medal in 1958. He was president of theLondon Mathematical Society from 1941 to 1943 and was awarded theDe Morgan Medal in 1938 and theSenior Berwick Prize in 1960.

Littlewood died on 6 September 1977.

Most of Littlewood's work was in the field ofmathematical analysis. He began research under the supervision ofErnest William Barnes, who suggested that he attempt to prove theRiemann hypothesis: Littlewood showed that if the Riemann hypothesis is true, then theprime 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.^[7][8]

In 1914, Littlewood published his first result in the field ofanalytic number theory concerning the error term of theprime-counting function. Ifπ(x) denotes the number of primes upx, then theprime number theorem implies thatπ(x) ~ Li(x),where${\displaystyle \mathrm{Li}(x)={\int }_2^x\frac{\mathrm{d}t}{\log t}}$  is known as theEulerian logarithmic integral.Numerical evidence seemed to suggest thatπ(x) < Li(x) for allx. Littlewood, however proved^[9] that the differenceπ(x) − Li(x) changes sign infinitely often.

Littlewood collaborated for many years withG. H. Hardy. Together they devised thefirst Hardy–Littlewood conjecture, a strong form of thetwin prime conjecture, and thesecond Hardy–Littlewood conjecture.

He also, with Hardy, identified the work of the Indian mathematicianSrinivasa Ramanujan as that of a genius and supported him in travelling from India to work at Cambridge.^[10] A self-taught mathematician, Ramanujan later became aFellow of the Royal Society, Fellow ofTrinity College, Cambridge, and widely recognised as on a par with other geniuses such asEuler andJacobi.^[11]

In the late 1930s, as the prospect of war loomed, theDepartment of Scientific and Industrial Research sought the interest of pure mathematicians in the properties ofnon linear differential equations that were needed by radio engineers and scientists. The problems appealed to Littlewood andMary Cartwright, and they worked on them independently during the next 20 years.^[12]

The problems that Littlewood and Cartwright worked on concerneddifferential equations arising out of early research onradar: their work foreshadowed the modern theory of dynamical systems.Littlewood's 4/3 inequality on bilinear forms was a forerunner of the laterGrothendiecktensor norm theory.

During theGreat War, Littlewood served in theRoyal Garrison Artillery as a second lieutenant. He made highly significant contributions in the field of ballistics.^[13][14]

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

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

Among his PhD students wereSarvadaman Chowla,Harold Davenport, andDonald 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 toLittlewood's conjecture).

Littlewood's collaborative work, carried out by correspondence, covered fields inDiophantine approximation andWaring's problem, in particular. In his other work, he collaborated withRaymond Paley onLittlewood–Paley theory inFourier theory, and withCyril Offord in combinatorial work on random sums, in developments that opened up fields that are still intensively studied.

In a 1947 lecture, the Danish mathematicianHarald 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.' "^{[15]: xxvii}

The German mathematicianEdmund 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".^[16] 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 aboutNorbert Wiener, who vehemently denied it in his autobiography.^[17]

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

John Littlewood is depicted in two films covering the life of Ramanujan –Ramanujan in 2014 portrayed byMichael Lieber andThe Man Who Knew Infinity in 2015 portrayed byToby Jones.

• 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
• Hardy–Littlewood definition
• Hardy–Littlewood inequality
• Hardy–Littlewood maximal function
• Hardy–Littlewood zeta function conjectures
• Hardy–Littlewood tauberian theorem
• First Hardy–Littlewood conjecture
• Second Hardy–Littlewood conjecture
• Ross–Littlewood paradox
• Hadamard three-circle theorem
• Skewes's number

• Burkill, J. C. (1978). "John Edensor Littlewood. 9 June 1885–6 September 1977".Biographical Memoirs of Fellows of the Royal Society.24:322–326.doi:10.1098/rsbm.1978.0010.JSTOR 769763.S2CID 119754496.
• Bateman, P; Diamond, P (1978). "John E. Littlewood (1885–1977) An Informal Obituary".The Mathematical Intelligencer.1 (1):28–33.doi:10.1007/BF03023041.S2CID 122107252.

• Littlewood's Miscellany, edited byB. Bollobás,Cambridge University Press; 1986.ISBN 0-521-33702-X (alternative title forA Mathematician's Miscellany)

• O'Connor, John J.;Robertson, Edmund F.,"John Edensor Littlewood",MacTutor History of Mathematics Archive,University of St Andrews
• John Edensor Littlewood at theMathematics Genealogy Project
• Papers of Littlewood on Number Theory
• A Mathematicians Miscellany