# John Edensor Littlewood

### Born: 9 June 1885 in Rochester, Kent, England

Died: 6 September 1977 in Cambridge, England

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**J E Littlewood**'s parents were Edward Thornton Littlewood and Sylvia Maud Ackland. A very few close friends called him Jack when he was elderly, but otherwise he would be addressed as Littlewood, which was not unusual between friends at that time. Hardy with whom he had a close collaboration for many years would write to him as "Dear L".

His father Edward Thornton Littlewood was also a mathematician and was Ninth Wrangler in the Mathematical Tripos at Cambridge in 1882, three years before his eldest son John Edensor was born. Edward and Sylvia Littlewood went on to have three sons, their second being Martin Wentworth Littlewood, who went on to study medicine, and a third son who tragically died when he was eight years old by falling into a lake from a bridge. When John Edensor was seven years old his father had to make a choice between two offers he received, one of a Fellowship at Magdalene College, Cambridge, or the second the position of Headmaster at a new school in Wynberg in South Africa. He chose the second and in 1892 the family sailed to South Africa.

Littlewood later wrote (see for example [7]):-

If the climate and scenery were superb, certainly the education that the young Littlewood had in South Africa was not. The quality of his school teachers was poor, and he was confused by the mathematics teaching that he received to the extent that he failed an arithmetic examination. He still did quite well and entered the University of Cape Town at a young age but again found that he was not able to benefit from teaching which was less than outstanding. His father soon realised that the University of Cape Town was not going to encourage his son's mathematical talents to blossom and Littlewood's parents made the decision to send their son back to England. We should mention that Littlewood, as we explain in more detail below, suffered from depression for most of his life, beginning while he was at school, and this may have contributed to his difficult days in the South African education system.Our lives would have been very different in Cambridge. ... the children of dons acquire an easy self-confidence ... As it was I had a very happy childhood among mountains, the ocean, and a beautiful climate.

In 1900 Littlewood, then aged 15, returned to England and entered St Paul's School in London. He was very fortunate to have there an outstanding teacher of mathematics Francis Macaulay [7]:-

While at St Paul's School in December 1902, Littlewood won a scholarship to Cambridge.This was education with a university atmosphere; at no school could Littlewood have acquired a better foundation of self-reliance, knowledge and judgement. He understood uniform convergence, and he could discriminate between basic ideas and tricks of manipulation.

Littlewood entered Trinity College Cambridge in October 1903. He wrote in [3]:-

His tutor at Trinity was Walter Rouse Ball, the author of the famous popular bookTo be in the running for Senior Wrangler one had to spend two-thirds of the time practising how to solve difficult problems against time ... I do not claim to have suffered high-souled frustration. I took things as they came; the game we were playing came easily to me, and I even felt a sort of satisfaction in successful craftsmanship.

*Mathematical Recreations and Essays*. After being equal first in Part I, he completed Part II in 1906 and began research under the direction of his tutor E W Barnes. Rapidly solving the first problem which Barnes gave him, Littlewood was next presented with the Riemann hypothesis as his next research problem by Barnes. Swinnerton-Dyer, in a memorial address at Trinity College after Littlewood's death, said:-

Littlewood never regretted having tackled the Riemann hypothesis, remarking that if one attempted a problem that was too difficult then one would always end up proving some interesting related results.It is an amazing illustration of the isolation and insularity of British mathematics at that time that Barnes should have thought it suitable for even the most brilliant research student, and that Littlewood should have tackled it without demur.

From 1907 to 1910 he lectured as Richardson Lecturer at the University of Manchester. He became a fellow of the Trinity College in 1908, winning a Smith's prize in that year, then returning to Trinity in 1910 to fill the position left vacant when A N Whitehead was essentially forced out of his job. Shortly after this, certainly by 1911 but some suggest in 1910, Littlewood began his famous collaboration with G H Hardy which we discuss in more detail below.

In World War I Littlewood served in the Royal Garrison Artillery. His contributions were highly significant and special allowances were made to keep him happy, such as letting him live with friends in London, and to carry an umbrella when in uniform! Littlewood himself described this war work in [14]. The result was that he improved the accuracy of anti-aircraft range tables and improved the formulae for finding the range, the time of flight and the angle of descent at the end of a trajectory with small elevation. E A Milne has described how Littlewood was able to discover techniques which greatly reduced the amount of work needed for making these accurate calculation of missile trajectories. Trials were conducted to see if the results of Littlewood's predictions held in practice and, Milne writes:-

Littlewood become Rouse Ball professor of mathematics in Cambridge in 1928. This chair had been founded by a benefaction from Walter Rouse Ball after his death in 1925, and Littlewood was its first occupant. It was a particularly appropriate choice, not only because of his outstanding mathematical contributions, but also since Littlewood had been tutored by Rouse Ball in his undergraduate days at Trinity. As Rouse Ball Professor, Littlewood could lecture on topics of his own choice and he no longer had to take part in routine teaching. It was an aspect which he enjoyed, delivering courses on his own wide areas of interest in analysis.... to the astonishment and joy of all concerned the observed positions of the shellbursts fell exactly on Littlewood's trajectories, at the correct time-markings, within very small errors of observation.

Almost all of Littlewood's mathematical research was in classical analysis, but in this area he looked at a remarkable range of subjects and he used an even broader range of techniques in proving his results. For 35 years he collaborated with G H Hardy working on the theory of series, the Riemann zeta function, inequalities, and the theory of functions. The collaboration led to a series of papers

*Partitio numerorum*using the Hardy-Littlewood-Ramanujan analytical method. During the years of this collaboration Littlewood was seldom seen outside Cambridge, in fact there were jokes around that he was the invention of Hardy. The real reason was rather a sad one, namely that Littlewood suffered from depression which we shall discuss more fully below. The rules that Hardy and Littlewood adopted for their collaboration were spelled out by Harald Bohr in a lecture which he gave in 1947 (see for example [11] where there is also an interesting discussion as to how far they stuck to their own rules and how far they ignored them):-

*When one wrote to the other, it was completely indifferent whether what they wrote was right or wrong*.

*When one received a letter from the other, he was under no obligation to read it, let alone answer it*.

*Although it did not really matter if they both simultaneously thought about the same detail, still it was preferable that they should not do so*.

*It was quite indifferent if one of them had not contributed the least bit to the contents of a paper under their common name*.

In the late 1930's the Department of Scientific and Industrial Research tried to interest pure mathematicians in nonlinear differential equations which were important for radio engineers and scientists because they described the behaviour of electric circuits. The impending war motivated this interest and in 1938 the Radio Research Board asked British pure mathematicians for help in dealing with certain types of nonlinear differential equations arising in radio engineering. Littlewood, working jointly with Mary Cartwright, spent 20 years working on equations of this type such as van der Pol's equation. McMurran and Tattersall discuss this collaboration in [16], and in particular the work on van der Pol's equation is discussed in [17]. They write:-

Littlewood was elected a Fellow of the Royal Society in 1915. He received the Royal Medal of the Society in 1929, and the Sylvester Medal of the Society in 1943:-Van der Pol's experiments with nonlinear oscillators during the1920s and1930s stimulated mathematical interest in nonlinear differential equations arising in radio research. The problems caught the attention of British mathematicians M L Cartwright and J E Littlewood, initiating a collaboration that lasted more than ten years. Cartwright and Littlewood's analysis of the van der Pol equation and its generalizations led them to explore some interesting topological methods, including the development of a fixed-point theorem for continua invariant under a homeomorphism of the plane. They were among the earliest mathematicians to apply Poincare's transformation theory to the analysis of dissipative systems. Their research is among the earliest in large parameter theory and played a role in the development of the modern theory of dynamical systems and chaos theory.

He also received the Copley Medal of the Society in 1958:-... in recognition of his mathematical discoveries and supreme insight in the analytic theory of numbers. ... Littlewood, on Hardy's own estimate, is the finest mathematician he has ever known. 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.

As we mentioned on a couple of occasions above, one problem which Littlewood had to cope with throughout his life was depression. It had affected him from his school days and continued to trouble him throughout his career. There is no evidence that it in any way impaired his ability to do mathematics, indeed it is hard to see how it could have done so given what he achieved, yet it certainly did much to spoil his social life for it had the effect of making him withdraw into himself. Although his mathematical research did not seem to suffer, it certainly stopped him from undertaking other mathematical activities which involved meeting people. For example he was able to say late in his life that he had never been secretary or chairman of any committee. In fact although he was President of the London Mathematical Society in 1941-43 he never chaired a meeting, the vice-President taking the chair throughout his period of office.... in recognition of his distinguished contributions to many branches of analysis, including Tauberian theory, the Riemann zeta-function, and non-linear differential equations.

After he retired he did receive treatment for his depression which was quite effective, and after 1957 he felt more able to accept invitations to visit colleagues. Indeed he made many visits to the United States in the ten years following 1957. We note that his problems of meeting people did not extend to those whom he knew well, for example he fully participated in the conversations at dinner in Trinity and greatly enjoyed the company in the familiar surroundings.

We should end this biography by saying something about Littlewood's interests outside mathematics. He [7]:-

His [2]:-... was slightly below average in height, strongly built and agile. At school he had been one of the best gymnasts and a hard hitting batsman. ... he was a keen follower of ball games and watched cricket at Fenner's on summer afternoons. He had an intense interest in music(classical - particularly Bach, Beethoven and Mozart); he had taught himself as an adult to play the piano ... He was best known to unmathematical undergraduates at Trinity for his skill in circling the seven yards of a pillar of the Library on the narrow ledge of its base and for his daily walk across the court to the baths, with a towel but no shirt. On most days he walked many miles in the country.

He continued to produce excellent mathematical results for many years after he retired and even when ninety years old he was still sharp and able to come up with new deep ideas. In 1970 he published a paper in which he wrote that he had solved a problem which:-... muscular strength and quickness of reaction made for success in rock climbing and skiing, and he spent many holidays in Cornwall, Scotland and Switzerland.

In addition to the honours which we have mentioned above, Littlewood was honoured with election to the Akademie der Wissenschaften in Göttingen in 1925, the Swedish Academy in 1948, the Royal Danish Academy also in 1948, and the Dutch Academy in 1950. In 1957 he was elected to the Paris Académie des Sciences to replace Fréchet. We mentioned above that Littlewood was President of the London Mathematical Society, but that Society also honoured him with its De Morgan Medal in 1938 and its Senior Berwick Prize in 1960 for two papers which he wrote on celestial mechanics.... raised difficulties which defeated me for some time. I have now overcome them.

Finally we note that he received honorary degrees from the University of Liverpool in 1928, the University of St Andrews in 1936 and the University of Cambridge in 1965.

**Article by:** *J J O'Connor* and *E F Robertson*

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**List of References** (19 books/articles)

**Some Quotations**(24)

**Mathematicians born in the same country**

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**Honours awarded to J E Littlewood**

(Click the link below for those honoured in this way)

1. | Fellow of the Royal Society | 1916 | |

2. | Royal Society Copley Medal | 1929 | |

3. | Royal Society Royal Medal | 1929 | |

4. | LMS De Morgan Medal | 1938 | |

5. | Royal Society Sylvester Medal | 1943 | |

6. | LMS President | 1941 - 1943 | |

7. | LMS Berwick Prize winner | 1960 | |

8. | Popular biographies list | Number 186 |

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