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TRIBUTE
Contemporary collaborator
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December 22 is Ramanujan's 116th birth anniversary. KRISHNASWAMI ALLADI pays tribute by describing the life and contributions of eminent British mathematician Littlewood, a contemporary of Ramanujan.
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Significant partnerships: Ramanujam.
J. E. LITTLEWOOD, an outstanding analyst and number theorist, was an eminent British mathematician of the 20th century. He was a contemporary of Ramanujan. In addition to his own fundamental contributions, Littlewood is equally famous for his collaboration with G. H. Hardy, Ramanujan's mentor. Indeed, the Hardy-Littlewood collaboration is considered to be the most remarkable and successful partnership in mathematical history, both in terms of its excellence and longevity. The Hardy-Ramanujan collaboration, even though it was brief, was equally significant. Although Littlewood was Ramanujan's contemporary, he did not collaborate with the Indian genius. But owing to closeness with Hardy, and because several mathematical problems were of common interest, there are connections between Littlewood and Ramanujan.
J. E. Littlewood was born on 9 June 1885 in Rochester in Kent as the eldest son to his parents. His father who was himself a Wrangler in the 1882 Cambridge Mathematical Tripos, turned down a Fellowship at Oxford, and opted to become a headmaster of a new school near Cape Town. Thus Littlewood lived in South Africa from 1892 to 1900.
Only after he returned to England in 1900 was Littlewood exposed to good mathematics by the distinguished algebraist F.S. Macauley whose approach was not to spoonfeed his students, but to make them work independently. Macauley's dictum was "Try a hard problem. You may not solve it, but you will prove something else." Littlewood later adopted this philosophy when training his own students.
Littlewood obtained an Entrance Scholarship to Trinity College at Cambridge University where he began residence in October 1903. He received training for the Tripos under R. A. Herman, described by Hardy as "the mildest of the most ferocious of the Huns". Littlewood emerged as the Senior Wrangler in Part I of the Tripos, yet he felt he wasted time gaining expertise in solving exceedingly difficult problems against the clock. After a vacation in 1906, Littlewood began research at Cambridge under the direction of E.W. Barnes who gave him a hard problem on integral functions of zero order. Luckily, Littlewood cracked this one and came out with a substantial paper. Next, Barnes asked Littlewood to work on the Riemann Hypothesis, the most celebrated problem in analytic number theory. Littlewood, who liked challenges, took this up, and managed to obtain some worthwhile results, which he wrote up as a dissertation for a Trinity Junior Fellowship that he received, but only the following year. Meanwhile, he was awarded the Smith's Prize, and received a lectureship at the University of Manchester, which he accepted. After going to Manchester, he realised that the work load there was too heavy, and so he returned to Cambridge in 1910 as College Lecturer at Trinity, succeeding Alfred North Whitehead whose lectures had inspired him as an undergraduate. The next year, Littlewood proved a profound converse of a famous theorem of Norwegian mathematician Abel on the summation of series.
Soon after he returned to Cambridge Littlewood began his collaboration with Hardy, which lasted about 35 years until Hardy's death in 1947. Their personalities and styles were very different. Rather than being similar, what was important here was that their working methods and ideas complemented each other perfectly to provide a broader scope and impact. Littlewood was considered the more imaginative of the two, was immensely powerful in analytical methods, and enjoyed challenging problems. Hardy was the finest mathematical craftsmen, had an eye for beautiful mathematical structure, and a superb writing style. In writing a joint paper, Littlewood would provide a logical skeleton in shorthand and Hardy always wrote the final draft.
To ensure that their partnership would not infringe on their independence and freedom, and to maintain their friendship and mutual respect, they formulated the following four axioms for their collaboration:
When one wrote to the other, it should not matter whether what was written was right or wrong.
When one received a letter from the other, he was under no obligation whatsoever to read it, let alone answer it.
Although it did not matter if they both thought about the same detail, still, it was preferable that they should not do so.
It did not matter at all if one of them had not contributed the least bit to the contents of a paper under their common name.
The great Danish mathematician Harald Bohr (brother of the Nobel Laureate physicist Neils Bohr) remarked "Seldom — or never — was such an important and harmonious collaboration founded on such apparently negative axioms."
In the post-Newtonian era, although there were eminent mathematicians in England, British mathematics took a back seat compared to the achievements made in mainland Europe. Hardy and Littlewood, who dominated the scene in England in the first half of the 20th Century, were attempting to resurrect the glory of British mathematics.
Naturally, as someone who was so close to Hardy, and as one working on problems at the interface between Number Theory and Analysis, there are strong links between Littlewood and Ramanujan.
K.R. JAWAHARR
Connections with the Indian genius: Littlewood.
When Ramanujan's letters arrived from India, the baffled Hardy consulted Littlewood for estimation. It was Littlewood who told Hardy that Ramanujan was in the class of Euler and Jacobi.
After Hardy and Ramanujan wrote their path breaking paper in 1917 in which they applied the Farey dissection of the unit circle to obtain an asymptotic formula for the number of partitions of an integer, Hardy and Littlewood, over the next few decades developed this "circle method" into a powerful tool of wide applicability encompassing various additive questions in number theory. In this regard, Hardy and Littlewood wrote a series of influential papers under the title "Some problems in partitio numerorum".
In discussing the validity of various astonishing claims made by Ramanujan, Hardy pointed out that in the study of prime numbers, Ramanujan (and other leading mathematicians) was sometimes misled by intuition. For example, in connection with the distribution of prime numbers, there were many who believed that $\pi(x)$, the number of prime numbers not exceeding $x$, was smaller than $\ell i(x)$, called the logarithmic integral of $x$, which is the integral of the reciprocal of $log (x)$ from 2 to $x$. All numerical evidence pointed to this. It was Littlewood who showed that in fact the difference $\pi (x) - \ell i(x)$ changes sign infinitely often. Littlewood's method did not yield the first value of $x$ when the sign change would take place. Subsequently the British mathematician Skewes showed that the first such sign change would take place definitely before $$ 10{circ}{lcub}10{circ}{lcub}10{circ}{lcub}34{rcub}{rcub}{rcub}.
$$ Hardy humorously remarked that this must be the largest number to have ever served a definite purpose in mathematics. He also notes that this number is much larger than the number of protons in the universe, which is of the order of $10{circ}{lcub}80{rcub}$.
During the First World War, Littlewood became a Second Lieutenant in the Royal Garrison Artillery and therefore was away from Cambridge. Thus he did not collaborate with Ramanujan. During this period Littlewood improved methods for calculating trajectories of anti-aircraft missiles. Hardy, who on the other hand was as ardent pacifist, remained in Cambridge and therefore could work with Ramanujan. Although he did not collaborate with Ramanujan, Littlewood was instrumental in working with Hardy to get Ramanujan elected as Fellow of the Royal Society (FRS).
In a long and distinguished career, Littlewood received various coveted honours and recognition in a steady stream. Being younger than Hardy who was considered more senior and prominent, recognitions for Littlewood followed those for Hardy. Littlewood succeeded Hardy as the Cayley Lecturer in Cambridge. Littlewood was elected Fellow of the Royal Society in 1916. He received the Royal Medal of the Society in 1929, nine years after Hardy. In 1928 Littlewood was given the newly formed Rouse Ball Professorship in Mathematics at Cambridge University, a very appropriate appointment because he had been one of Rouse Ball's favourite pupils in Trinity. That same year Littlewood received his first Honorary Doctorate - from the University of Liverpool.
Littlewood succeeded Hardy as president of the London Mathematical Society during 1941-43. Like Hardy he received the De Morgan Medal of the Society in 1938. Littlewood was also awarded the Sylvester Medal in 1943, and the Copley Medal in 1953.
Littlewood retired in 1950 at the statutory age of 65. He suffered for many years from a nervous disorder which in 1960 was cured by a brilliant neurologist by the discovery of a new drug. This gave Littlewood a new lease of life and at the age of 75 he started visiting the United States, accepting lecture engagements. He continued his mathematical researches well into his eighties. He died on 7 September 1977 at the age of 92.
G. H. Hardy remarked, that no other mathematician can boast of having had the privilege of collaborating with Littlewood and Ramanujan on something like equal terms. Whereas the Hardy-Littlewood collaboration achieved so much over an extended period, the Hardy-Ramanujan partnership produced spectacular results in a remarkably short span of time. On the Hardy-Ramanujan asymptotic formula for partitions Littlewood commented, "We owe the theorem to a singularly happy collaboration of two men, of quite unlike gifts, in which each contributed the best, most characteristic, and most fortunate work that was in him." In ranking mathematicians on the basis of pure talent, Hardy gave Ramanujan the perfect score of 100, the great German mathematician Hilbert the score of 80, himself the score of only 25, and Littlewood the score of 30. These ratings of mathematicians reflect Hardy's humility, his respect for Littlewood as the more brilliant of the two, and his highest admiration for the genius of Ramanujan.
The writer is with the University of Florida, Gainesville.
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