Littlewood, John Edensor

(1885-1977), mathematician

by Béla Bollobás

© Oxford University Press 2004 All rights reserved

Littlewood, John Edensor (1885-1977), mathematician, was born on 9 June 1885, at 4 Clevedon Terrace, Roebuck Lane, Rochester, Kent, the eldest of three children of Edward Thornton Littlewood (1859-1941), schoolmaster, and Sylvia Maud Ackland. Littlewood's mother was part Irish, but his father was from old English stock--a Littlewood fought at Agincourt. Littlewood's grandfather, the writer the Revd William Edensor Littlewood (1831-1886), had been educated at Pembroke College, Cambridge, and graduated thirty-fifth wrangler in the mathematical tripos. Littlewood's father went to Peterhouse, Cambridge, and was ninth wrangler; in 1892 he accepted the headmastership of a school at Wynberg, near Cape Town. Littlewood spent eight years of his childhood in South Africa, attending school and some classes at Cape Town University. Then, at the age of fourteen, he was sent by his father to St Paul's School, London where, under F. S. Macaulay, he acquired an excellent grounding in mathematics.

In 1903 Littlewood proceeded to Trinity College, Cambridge. There was cut-throat competition to do well in the Order of Merit (abolished in 1910), but, as he wrote later, 'the game we were playing came easily to me' (Littlewood, Littlewood's Miscellany, 84), and in 1905 he was senior wrangler, bracketed with J. Mercer who had previously graduated at Manchester. In his third year he took part two of the mathematical tripos: this was much less of a sporting event and contained much genuine mathematics.

In the long vacation of 1906 Littlewood started research under E. W. Barnes (later bishop of Birmingham). His first topic was integral functions of zero order. Littlewood 'rather luckily struck oil at once by switching to more "elementary" methods' (Littlewood, Littlewood's Miscellany, 88). The next problem was rather different: 'prove the Riemann hypothesis'--the 1860 conjecture of the great German mathematician Bernhard Riemann that all 'non-trivial' zeros of the [Riemann] zeta-function are on the 'critical line', an assertion closely related to the distribution of prime numbers. Most mathematicians now consider the Riemann hypothesis the most famous problem in mathematics, whose solution seems far away despite many powerful attacks. It is a clear proof of the isolated and backward state of British mathematics at the beginning of the twentieth century that Barnes should have thought it suitable for a research student. Nevertheless, Littlewood attacked the hypothesis and even benefited from it. This led him to try hard problems because if one fails to solve them something else will be proved.

Between 1907 and 1910 Littlewood was Richardson lecturer at Manchester University. In 1908 he won a Smith's prize, and was elected into a prize (research) fellowship in Trinity College, returning to Cambridge two years later to succeed A. N. Whitehead on the Trinity mathematics staff. His reading at this time of Landau's newly published Primzahlen marked the end of his education. Soon he wrote his first important paper (on the converse of Abel's theorem) and started his collaboration with his Trinity colleague G. H. Hardy (1877-1947). The pattern was set for his professional life: in 1912 he moved into a spacious set of rooms in Nevile's Court; with the exception of the First World War, he occupied these rooms for the next sixty-five years, until his death.

The Hardy-Littlewood collaboration, the greatest collaboration in all of mathematics, continued for thirty-five years, their last paper being published a year after Hardy's death. In a hundred joint works they made fundamental contributions to summability, inequalities, function theory, Diophantine approximation, Fourier series, and, above all, number theory. They worked on the theory of the Riemann zeta-function, proving the first major results about the zeros of the zeta-function on the critical line. In their series of papers entitled Partitio numerorum they developed the powerful Hardy-Littlewood circle method to overcome formidable technical difficulties, so creating a research area that is still very active. Hardy and Littlewood also greatly extended earlier results of David Hilbert concerning Waring's problem. Landau expressed a widely held view when he said that 'The mathematician Hardy-Littlewood was the best in the world, with Littlewood the more original genius, and Hardy the better journalist'. Their habit of communicating via letters meant that, when Hardy left Cambridge during 1919-1931 for a chair at Oxford, their collaboration rose to new heights rather than withered. In 1926 they wrote a highly original paper on rearrangements of sequences, and in 1930 they proved their very influential 'maximal theorem'.

As related by their Danish colleague Harald Bohr:

when they planned and began their far-reaching and intensive team work, they still had some misgivings about it because they feared that it might encroach on their personal freedom, so vitally important to them. Therefore ... they amused themselves by formulating four so-called 'axioms' for their mutual collaboration. The first of them said that, when one wrote to the other ... it was completely indifferent whether what they wrote was right or wrong. ... The second axiom was to the effect that, when one received a letter from the other, he was under no obligation whatsoever to read it, let alone to answer it. ... And they really observed this axiom to the fullest extent. When Hardy once stayed with me in Copenhagen, thick mathematical letters arrived from Littlewood, who was obviously very much in the mood for work, and I have seen Hardy calmly throw the letters into a corner of the room, saying: 'I suppose I shall want to read them some day'. The third axiom was to the effect that, although it did not really matter if they both thought about the same detail, still, it was preferable that they should not do so. And, finally, the fourth, and perhaps most important axiom stated that it was quite indifferent if one of them had not contributed the least bit to the contents of a paper under their common name; otherwise there would constantly arise quarrels and difficulties in that now one, and now the other, would oppose being named co-author. I think one may safely say that seldom--or ever--was such an important and harmonious collaboration founded on such apparently negative axioms. (Littlewood, Littlewood's Miscellany, 9-11)
In 1914 Littlewood published his famous result concerning the error term in the prime number theorem, which asserts that the 'logarithmic integral' li(x) is a good approximation of z(x), the number of primes up to the number x. Empirical evidence suggests that z(x) is less than li(x), but Littlewood proved the surprising result that z(x) - li(x) changes sign infinitely often, giving bounds for the difference that are essentially best to this day. This result was instrumental in his election to the Royal Society in 1916.

From 1914 to 1918 Littlewood served in the Royal Garrison Artillery, working on the somewhat utilitarian mathematics of gunnery. It was at this time that the Indian genius Srinivasa Ramanujan (1877-1920) went to Cambridge on the invitation of Hardy. Littlewood's absence during the war meant that his contact with Ramanujan was never more than slight. In 1928 Littlewood was elected to the newly established Rouse Ball chair of mathematics, founded by a benefaction of his former tutor. From then on, he had no college teaching and could give lectures on topics of his choice. His reputation was well established and the quality of his mathematical output was very high. In 1934 he published Inequalities with Hardy and G. Pólya: it became an instant bestseller and is still popular. From 1928 he held a weekly conversation class in his rooms for advanced students (he had close on thirty research students in all); on Hardy's return from Oxford in 1931 this metamorphosed into a larger gathering run by Hardy, and became known as the 'Hardy-Littlewood conversation class at which Littlewood is never present'.

It was not only with Hardy that Littlewood did much important collaborative work. With his outstanding student R. C. A. Paley he developed powerful ideas to tackle problems in harmonic analysis, but this highly successful collaboration was cut short by Paley's untimely death. From the late 1930s Littlewood and A. C. Offord investigated the distribution of zeros of random polynomials and entire functions. Among many other results, they discovered the curious phenomenon of 'pits effect' that Littlewood continued to study until 1970. Their original work on random algebraic objects was decades ahead of its time.

For the last forty years of his life, from just before the Second World War, Littlewood took a great interest in ordinary differential equations. It began when a Radio Research Board memorandum asked for help with differential equations connected to radio oscillators and thermionic valves (vacuum tubes). M. L. Cartwright took up the challenge and fired Littlewood's own interest in the problems. They began to study the behaviour of the stable periodic solutions of Balthasar van der Pol's equation. Littlewood later recalled that for something to do they went on and on with no prospect of 'results', but suddenly the entire vista of the dramatic fine structure of solutions stared them in the face. The twelve-year collaboration of Littlewood and Cartwright produced three joint papers and several others published under one or other name only, including Littlewood's enormous 1957 paper that he called 'the monster', saying of it 'I should never have read it had I not written it myself' (Littlewood, Littlewood's Miscellany, 16). Littlewood and Cartwright were among the first to recognize the combined power of topological and analytical methods to tackle deep problems in differential equations, discovering the phenomenon now known as 'chaos', although this had already been hinted at by Poincaré.

Littlewood abhorred administration and official position of any kind--later in life he was proud to say that he had never been chairman or secretary of any body. However, he was president of the London Mathematical Society from 1941 to 1943, and was also happy to be a foreign member of various academies and to receive honorary doctorates from several universities, including one from Cambridge in 1965. He received the royal (1929), Sylvester (1943), and Copley (1958) medals of the Royal Society, and the De Morgan medal (1938) and senior Berwick prize (1960) of the London Mathematical Society. He retired from his chair in 1950, but he continued publishing and went on several lecture tours to the USA. In 1953 he published A Mathematician's Miscellany, a delightful book of essays about his education, mathematics, and academic life; twenty-six years later a considerably expanded version of it was published as Littlewood's Miscellany.

Littlewood was below average in height but was strongly built and athletic. In school he was a good gymnast and batsman; later he was a keen swimmer and rock climber. He took up skiing in Switzerland in 1924, and went on skiing well into his seventies. He must have thought of himself when he wrote: 'Mathematics is very hard work, and dons tend to be above the average in health and vigour. Below a certain threshold a man cracks up, but above it hard work makes for health and vigour' (Littlewood, Littlewood's Miscellany, 195). However, he suffered from bouts of depression, occasionally in his youth but more intensely in later years. It was an enormous relief to him when in 1960 his illness responded to newly discovered drugs.

Littlewood spent most of his summer vacations on the Cornish coast, swimming, walking, and climbing. He never married, but had a son, Philip Streatfeild, and a daughter, Ann Streatfeild, with whom he spent all his time away from Cambridge. He read voraciously, not only classical literature but also science fiction and thrillers, and he listened to records of Bach, Beethoven, and Mozart many hours a day. His range of conversation was exceptional and his presence enlivened every gathering. He died on 6 September 1977 in the Evelyn Nursing Home, Cambridge, having suffered heart failure after a bad fall in his room.

Littlewood was undoubtedly one of the greatest English mathematicians of the first half of the twentieth century--together with Hardy he created a school of pure mathematics second to none. Hardy considered him the finest mathematician he ever knew: 'He was the man most likely to storm and smash a really deep and formidable problem; there was no one else who could command such a combination of insight, technique and power' (Littlewood, Littlewood's Miscellany, 22).

BÉLA BOLLOBÁS

Sources  
M. L. Cartwright, 'Later Hardy and Littlewood manuscripts', Bulletin of the London Mathematical Society, 17 (1985), 318-390
Collected papers of G. H. Hardy: including joint papers with J. E. Littlewood and others, ed. London Mathematical Society, 7 vols. (1966-79)
G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities (1988)
E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, 1-2 (1909)
J. E. Littlewood, A mathematician's miscellany (1953)
J. E. Littlewood, Collected papers of J. E. Littlewood, 2 vols. (1982)
J. E. Littlewood, Littlewood's miscellany, ed. B. Bollobás (1986)
S. L. McMurran and J. J. Tattersall, 'The mathematical collaboration of M. L. Cartwright and J. E. Littlewood', American Mathematical Monthly, 103 (1996), 833-45
R. C. Vaughan, The Hardy-Littlewood method, 2nd edn (1997)
H. Bohr, Collected mathematical works, ed. E. Folner and B. Hessen, 1 (1952), xxvii-xxviii
b. cert.
d. cert.
J. C. Burkill, Memoirs FRS, 24 (1978), 323-67

Archives  
Bodl. Oxf., autobiographical papers
CUL, papers on Waring's problem
Trinity Cam., papers |  McMaster University, Hamilton, Ontario, corresp. with Bertrand Russell
Trinity Cam., corresp. with A. E. Ingham

Likenesses  
W. Stoneman, photograph, 1932, NPG
R. Brill, pen-and-ink and sepia drawing, 1933, Trinity Cam.
G. Bollobás, bronze bust, 1973, Trinity Cam.
G. Bollobás, bust, Institution of Mathematics and its Applications, Southend-on-Sea, Essex
photograph, repro. in Burkill, Memoirs FRS, facing p. 323
photograph, repro. in Littlewood, Collected papers

Wealth at death  
£16,895: probate, 12 Dec 1977, CGPLA Eng. & Wales


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